G 4 Flux, Algebraic Cycles and Complex Structure Moduli Stabilization
aa r X i v : . [ h e p - t h ] S e p G Flux, Algebraic Cycles andComplex Structure Moduli Stabilization
A. P. Braun a and R. Valandro ba Department of Mathematical Sciences, Durham University, Lower Mountjoy, StocktonRd, Durham DH1 3LE, UK b Dipartimento di Fisica, Università di Trieste, Strada Costiera 11, I-34151 Trieste, Italyand INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy
Abstract
We construct G fluxes that stabilize all of the 426 complex structure moduliof the sextic Calabi-Yau fourfold at the Fermat point. Studying flux stabilizationusually requires solving Picard-Fuchs equations, which becomes unfeasible for modelswith many moduli. Here, we instead start by considering a specific point in thecomplex structure moduli space, and look for a flux that fixes us there. We showhow to construct such fluxes by using algebraic cycles and analyze flat directions.This is discussed in detail for the sextic Calabi-Yau fourfold at the Fermat point,and we observe that there appears to be tension between M2-tadpole cancellationand the requirement of stabilizing all moduli. Finally, we apply our results to showthat even though symmetric fluxes allow to automatically solve most of the F-termequations, they typically lead to flat directions. ontents Introduction
M-theory compactified on a Calabi-Yau (CY) fourfold X has h , ( X ) complex structuremoduli, which can be thought of as variations of the holomorphic top form Ω . In suchmodels, one can include four-form fluxes G as part of the background, which preserve theCalabi-Yau metric up to warping [1]. Such fluxes give a potential to the complex structuremoduli at tree level, which can be expressed in the resulting three-dimensional N = 2 theory in terms of the Gukov-Vafa-Witten (GVW) superpotential [2] W GVW = Z X G ∧ Ω . (1.1)The minima of the induced scalar potential are solutions of the F-term equations D I W =0 , I = 1 , ..., h , ( X ) . They are supersymmetric Minkowski vacua if furthermore W GVW = 0 .This implies that the complex structure must be such that G ∈ H , ( X ) . It is commonlybelieved that a typical G flux fixes all of the complex structure moduli. The argumentfor this is simple: there are as many constraints as there are complex structure moduli.The implicit assumption which enters this argument is that each of the F-term equationsis linearly independent, which is expected to hold for a ‘generic’ choice of G .As a consequence of flux quantization [3], which says that G + c ( X )2 ∈ H ( X, Z ) ,sensible choices of G form a lattice, which begs the questions what precisely is meantby a ‘generic’ choice of flux in this context. Complicating matters even more, there is theconsistency condition commonly refereed to as M -tadpole cancellation [1], which boundsthe length squared of possible flux choices from above. Although it is always possible tofind lattice vectors such that all F-term equations become linearly independent, this mightrequire to pick lattice sites which are far away from the origin and hence too long to satisfythe bound imposed by the M -tadpole. The relevant question is hence: ‘is there a choiceof flux such that all F-term equations are independent and the bound imposed by M -tadpole cancellation is satisfied ?’.This is a difficult question to study in general, and it may well be that the tadpoleconstraint has a strong selective power. This observation becomes particularly interestingwhen the fourfold X is elliptically fibered and used as an F-theory background. In suchcompactifications, the four-dimensional gauge sector is engineered by appropriate singu-larities of X , and (part of) the complex structure moduli space of X corresponds e.g.to adjoint Higgs fields. Complex structure moduli that do not receive a potential from(1.1) hence give rise to flat directions in the gauge sector, and the inability to stabilize allcomplex structure moduli corresponds to such flat directions inevitably being present .On the other hand, loci of enhanced gauge symmetry are typically at very high codimen-sion in the moduli space [5–7] and it would be a fascinating scenario if the consistencyconditions only allowed fluxes that would select such loci for us [5, 8]. There is also the possibility that the F-term equations have no solutions, as explained in [4] forM-theory compactifications on K × K . Some of the open string moduli may sit in matter multiplets that must be massless at the classicallevel to be consistent with some phenomenological requirements. H ( X ) are integral, so that they can beused to define an appropriately quantized flux. One method to find such a basis is givenby mirror symmetry [9].The main motivation of the present work is to further explore an alternative approach.The crucial idea underlying this approach is as follows: at supersymmetric Minkowskivacua, the properly quantized flux must be an element of H , ( X ) ∩ H ( X, Z ) up toa shift c ( X ) . The group H , ( X ) ∩ H ( X, Z ) ≡ H Hodge ( X ) of Hodge cycles is notconstant throughout complex structure moduli space, but may be enhanced at specificloci, called Hodge loci. This is analogous to the enhancement of the Picard lattice of K3surfaces at Noether-Lefschetz loci. If we identify such a locus and switch on a flux whichis proportional to one of the Hodge cycles appearing there, the model cannot be deformedaway from this locus, as the flux is only of type (2 , on the Hodge locus, so that theassociated F-term equations are necessarily violated away from it. Instead of picking aflux in H ( X ) and asking where it drives the model, the strategy we want to use is to identify loci in the moduli space where supersymmetric fluxes are possible, and then ask ifwe can find a flux that traps it there . See [10] for a beautiful exposition of this idea.For K3 surfaces, the Torelli theorem implies that demanding for a single lattice vectorin H ( K , Z ) to be in H , ( K fixes one complex structure modulus. This is not truefor fourfolds, where the number of complex structure moduli we need to tune for a singleelement η in H ( X, Z ) to be in H , ( X ) depends on both X and η . Fixing all complexstructure moduli then corresponds to finding a so-called ‘general’ Hodge cycles for whichthe associated Hodge locus is just a point in the complex structure moduli space of X .If such a cycle furthermore satisfies the M -tadpole constraint (after adding the piece c ( X )2 ), there is a G -flux that stabilizes all complex structure moduli.In order to identify Hodge cycles and their Hodge loci, we will make use of algebraiccycles of complex dimension two. These are Poincaré dual to forms of Hodge type (2 , and it is not hard to find instances which only appear at special loci in the modulispace. Such an approach was followed in [11], and we will extend this work in severalaspects. In [11], the number of stabilized moduli was simply counted by working out howmany polynomial deformations are frozen by the existence of a given algebraic cycle. Asthis tacitly assumes the validity of a version of the Hodge conjecture, such a method isinsufficient for a reliably counting. This point which was adressed in [11] by using therelationship of complex structure moduli of F-Theory compactifications to open stringmoduli in IIB orientifolds, a way of reasoning that is not available for general M-Theorybackgrounds on Calabi-Yau fourfolds. Furthermore, one may consider fluxes which arePoincaré dual to some linear combination of algebraic cycles. In this instance, studyingpolynomial deformations is simply not powerful enough to detect all flat directions.Working with the sextic fourfold X at the Fermat point as a simple example, we show3ow to address both of these issues by directly evaluating the rank of the matrix G IJ ≡ Z X G ∧ D I D J Ω , (1.2)which counts the number of fixed complex structure moduli. The crucial ingredient neededto evaluate these integrals are the periods of variations of Ω over algebraic cycles, whichhave been computed for the sextic fourfold at the Fermat point in [12,13]. For the simplestclass of algebraic cycle we show how to recover the periods (up to overall normalization) byexploiting the automorphism group of X , and construct fluxes that stabilize all complexstructure moduli. These fluxes, however, significantly overshoot the tadpole constraintoriginating from the cancellation of M -brane charge. Although a computation that con-firms this in some form of generality is computationally too demanding to be within thescope of the present work, we take this as evidence for the tension between the M tadpolecancellation constraint and the desire to stabilize all complex structure moduli.As a further application we consider the interplay between fluxes and symmetries.In [33, 34] it was suggested to use fluxes respecting some symmetries of the complexstructure moduli space, in order to stabilize all moduli. The trick is that one needs to solveonly the F-term equations of the invariant moduli, as the (many) F-terms of non-invariantcomplex structure deformations automatically vanish at a symmetric point. However, theargument does not take into account possible flat directions. In fact, we show that suchflat directions are typically present in such setups. We give an example for the Fermatsextic fourfold. This shows that caution has to be taken in using the trick of turning onsymmetric fluxes to claim full complex structure moduli stabilization.After reviewing some aspects of flux compactification in M-Theory on Calabi-Yau four-folds in Section 2, we discuss algebraic cycles at the Fermat point of the sextic fourfoldin Section 3. In Section 4, we describe the middle cohomology of X , the span of algebraiccycles, and variations of the holomorphic top-form Ω using residues of holomorphic formswith poles on P . Some technical background on residues and rational forms are containedin an appendix. After introducing expressions for periods of residue forms on algebraiccycles, we apply these to several examples in Section 5, and give some estimates that quan-tify the tension between complete moduli stabilization and tadpole cancellation. Modulistabilization in the presence of fluxes respecting a symmetry is disussed in Section 6. Weclose with a discussion of open issues and future directions. In this paper we consider M-theory compactified on a CY fourfold X . The resulting lowenergy theory is a three dimensional (3d) N = 2 supergravity, i.e. a theory with four super-charges. The metric deformations preserving the Calabi-Yau condition are called metricmoduli and become massless scalars in the 3d theory. For CY fourfolds X , the metric mod-uli are encoded in the h , ( X ) periods of the Kähler form J and the h , ( X ) independentdeformations of the holomorphic (4 , -form Ω . These moduli are called Kähler moduli complex structure moduli , respectively. There are also h , ( X ) axionic moduli thatcome from the dimensional reduction of the eleven-dimensional (11d) sugra seven-form C (the dual of C ), which complexify the Kähler moduli .The dynamics of the moduli is determined by the Kähler potential K = K c.s. + K K , (2.1)with K c.s. = − ln (cid:18)Z X Ω ∧ ¯Ω (cid:19) and K K = − (cid:18) Z X J ∧ J ∧ J ∧ J (cid:19) . (2.2)One can switch on a non-zero vev for the four-form flux G = dC , that is quantizedaccording to G + c ( X )2 ∈ H ( X, Z ) . (2.3)where c ( X ) is the second Chern class of the tangent bundle of X .A non zero flux along internal directions generates a potential for the metric moduliafter compactification [14]. This can be understood from the 11d C kinetic term R G ∧∗ G , which depends on the metric through the Hodge star operator ∗ ). The minima ofthe supergravity scalar potential are given by the solutions of the following equations D I W = 0 I = 1 , ..., h , ∂ k ˜ W = 0 k = 1 , ..., h , (2.4)where W = Z X G ∧ Ω and ˜ W = Z X G ∧ J ∧ J . (2.5)Here W is the GVW superpotential [2] and D I = ∂ I + ∂ I K , with K the Kähler potential(2.1). The index I runs over the complex structure moduli, and the index k runs over theKähler moduli.These minima are at zero cosmological constant (i.e. they are Minkowski vacua). Theyare furthermore supersymmetric if the vev of W vanishes, i.e. W | min = 0 . This condi-tion together with (2.4) can be rephrased by saying that the four-form flux must lie in H , ( X ) , i.e. G must be a primitive four-form of Hodge type (2 , . We now explain this.The same can be done in the dual type IIB compactification on CY orientifolds [15], seealso [16] for an overview over the classic literature on the subject.We first explain why G is of Hodge type (2 , : • The condition W = 0 means Z X G ∧ Ω = 0; this implies that the (0 , component of G vanishes. Since G is real, also its (4 , component is zero. 5 The condition D I W = 0 means Z X G ∧ D I Ω = 0 ∀ I ; since the forms D I Ω give a basis of H , ( X ) [17, 18], the (1 , and (3 , componentsof G vanish.We then see that only the (2 , part of G survives.As regarding the primitivity condition, expand first the Kähler form J in a basis ofharmonic (1 , -forms ω k : J = t k ω k . t k are the h , ( X ) Kähler moduli. The condition ∂ k ˜ W = 0 means Z X G ∧ J ∧ ω k = 0 ∀ k that implies G ∧ J = 0 , i.e. G is a primitive form.When the flux, as required, belongs to H , ( X ) , then it is also self-dual , i.e. ∗ G = G .This, in particular, implies that the contribution of G to the M2-charge, i.e. Q flux M = 12 Z X G ∧ G , (2.6)is positive definite. In order to be possible to satisfy the M2-tadpole cancellation condition, Q flux M + N M = χ ( X )24 (2.7)without introducing anti-branes, Q flux M must be smaller than the contribution coming fromthe geometry, i.e. Q flux M ≤ χ ( X )24 .Let us now concentrate on the complex structure moduli. We choose a point in thecomplex structure moduli space that satisfies D I W = 0 and W = 0 . We take coordinates s I such that this point is at s = ( s , s , s , ... ) = 0 . The holomorphic (4 , -form at ageneric point is Ω( s ) and W ( s ) = R X G ∧ Ω( s ) . We then have D I W ( s ) | s =0 = 0 . (2.8)A flat direction of the potential is a curve s ( t ) in the moduli space passing through s = 0 at t = 0 that satisfies the minimum condition for all t in a neighborhood of t = 0 , i.e. D I W ( s ( t )) = 0 ∀ t . (2.9)Expanding around t = 0 and keeping the leading term at small t , one finds the infinitesimalexpression for (2.9), i.e. ˙ s J (0) ∂ J D I W (0) = 0 . (2.10)Notice that ∂ J D I W (0) = D J D I W (0) , since the two expressions differ by ( ∂ J K ) D I W (0) which vanishes because of (2.8). The vectors ˙ s J (0) solving (2.10) give the flat directions. In general D I W is not holomorphic, as there is the term ( ∂ I K ) W with ∂ I K non-holomorphic in thecomplex structure moduli chiral superfields. Hence one may expect in (2.10) a term involving ¯ ∂ J D I W .However the extra term is ( ∂ I ∂ ¯ J K ) W , which vanishes at s = 0 because of the W = 0 condition.
6e hence conclude that in order to have no flat directions at s = 0 we need that thematrix G IJ := D J D I W ( s ) | s =0 (2.11)has maximal rank. More generally, the rank of the matrix (2.11) counts how many complexstructure moduli are stabilized by G . This is called the codimension of the Hodge locusof G in math literature, see [19, 20] for a review.The Poincaré dual of an algebraic four-cycle is a four-form of type (2 , . When thefourfold is at a specific point in the complex structure moduli space, one may be able toconstruct explicit algebraic cycles, as we will do for the sextic fourfold. One can then usethem to define a choice of properly quantized flux that is a primitive (2 , -form at thatspecific point. The question we want to address here, is how many moduli are stabilizedonce such a flux is introduced: any deformation that breaks originates in a G flux notpurely of type (2 , is lifted by the flux potential.Let us come back to the GVW superpotential that generates the minima conditionfor the complex structure moduli. The part of the flux G that contributes to the super-potential, the F-term conditions and the stability condition is the one that has non-zerointersection with Ω( s ) and its derivatives. Here by ‘intersection’ we mean the productgiven by the inner form a · a ≡ R X a ∧ a . The holomorphic four-form and its derivativesdo not span the full middle cohomology H ( X ) , but only the primary horizontal sub-space [17, 18]. In contrast, forms of Hodge type (2 , defined by intersections of divisorslie in the primary vertical subspace, which is perpendicular to the primary horizontalsubspace . To study stabilization of complex structure moduli we hence need to consider G fluxes that lie in the horizontal subspace of H ( X ) (apart from the piece c ( X ) thatis forced on us by quantization). The algebraic cycles we consider here are exactly of thistype [11]. The manifold of interest to us in this paper is the sextic fourfold. A sextic fourfold at ageneric point in its moduli space is defined by the vanishing of a homogeneous polynomialof degree in P : X : x + x + x + x + x + x + X a c a Y x a i i = 0 (3.1)where a = ( a , · · · , a ) are integers such that P a i = 6 and the c a are complex coefficientsthat can be thought of as deformations of the complex structure. The topological numbersof X are h , ( X ) = 1 h , ( X ) = 0 h , ( X ) = 426 h , ( X ) = 1752 . (3.2)It follows that χ ( X ) = 2610 and b ( X ) = 1754 , b − ( X ) = 852 . In general, there can be directions in H , ( X ) which do not lie in either subspace [5]. h , ( X ) is generated by the restriction of the hyperplane class H of P , and any Kähler form on X is necessarily proportional to H . There is a uniquegenerator of the primary vertical subspace H , V ( X ) which is given by H · H ≡ H andwhich is always proportional to the square of the Kähler form.The orthogonal directions to H in H ( X ) are hence all primitive, i.e. h , prim ( X ) =1751 , and can be shown to all belong to the primary horizontal subspace H H ( X ) , whichhas dimension . The second Chern character of X is c ( X ) = 15 H . (3.3)The term c ( X ) is hence not integral, so that flux quantization forces us to include ahalf-integral flux proportional to H .For a typical choice of the c a , the only algebraic cycles contained in X are completeintersections of X with multiples of the hyperplane divisor in P . On X the classes of theseare proportional to H , so that the rank of H , ( X ) ∩ H ( X ) prim is zero. As H is neverprimitive, there are furthermore no supersymmetric fluxes along this direction. If we tunethe c a to special values, the situation changes and the rank of H , ( X ) ∩ H ( X , Z ) prim becomes non-zero.Let us hence make a specific choice and set all c a = 0 , which puts us on the Fermatpoint of the moduli space of the sextic. We will denote the sextic fourfold at the Fermatpoint by: X : x + x + x + x + x + x = 0 . (3.4)As the above equation describes a smooth submanifold of P , the topological numbers ofthe Fermat sextic are the same as those of X . Only the group H , ( X ) ∩ H ( X ) prim isdifferent from the case of a generic sextic fourfold: it has the maximal possible rank of . It is not hard to find the simplest type of algebraic cycle sitting inside X . Take e.g. x = αx , x = βx and x = γx for α = β = γ = − . In this case, these threeequations imply (3.4), so that they define a subvariety of complex codimension inside P , which is complex codimension in X .Using the large group of automorphisms of X , we can immediately write down thegeneral form of such cycles as C ℓ σ : x σ (0) = e iπ/ e iπℓ / x σ (1) x σ (2) = e iπ/ e iπℓ / x σ (3) x σ (4) = e iπ/ e iπℓ / x σ (5) (3.5) We will see this explicitly in Section 4.4. This can also be shown by computing that the dimensionof the primary vertical subspace of the mirror, h , V ( X ∨ ) = 1751 and using that primary vertical andhorizontal subspaces are swapped by the mirror map. This is also called the Gepner point (if one thinks in terms of the worldsheet CFT of strings propa-gating on X ) or the Brieskorn-Pham point (if one things in terms of singularity theory). ℓ i ∈ { , , , , , } specify which sixth root of unity we are using and σ is apermutation of { , , , , , } which specifies which coordinates are paired to form C ℓ σ .The existence of such algebraic cycles can also be inferred by writing the definingequation of X (3.4) in the following ‘factorized’ form Y ℓ =0 (cid:0) x σ (0) − e iπ/ e iπℓ / x σ (1) (cid:1) + Y ℓ =0 (cid:0) x σ (2) − e iπ/ e iπℓ / x σ (3) (cid:1) + Y ℓ =0 (cid:0) x σ (4) − e iπ/ e iπℓ / x σ (5) (cid:1) = 0 . (3.6)This gives a hint of how other instances of algebraic cycles can be found. Another factor-ization of the defining equation for X is: (cid:0) x + e iπk x + ix (cid:1) (cid:0) x + e iπk x − ix (cid:1) + Y m =0 (cid:16) x − / e iπk e πim x x (cid:17) + Y ℓ =0 (cid:0) x − e iπ/ e iπℓ/ x (cid:1) (3.7)up to permutation of the four coordinates and for k = 0 , . One then realizes the existenceof the algebraic cycles C kjmℓσ : x σ (0) + e iπk x σ (1) + ie iπj x σ (2) = 0 x σ (3) − / e iπk e πim x σ (0) x σ (1) = 0 x σ (4) − e iπ/ e iπℓ/ x σ (5) = 0 (3.8)for k, j ∈ Z / Z , m ∈ Z / Z and ℓ ∈ Z / Z . Note that x σ (4) and x σ (5) are paired in a similarway as before, whereas a more complicated factorization is used for the remaining fourcoordinates. These cycles are a lift of the cycles that were used to construct the Néron-Severi group of Fermat sextic surfaces in [21], where famously using only lines is no longersufficient [22].An example of a completely non-linear factorization of (3.4) is given by Y s =0 (cid:16) x + e πi ( k + s ) x + e πi ( k +2 s ) x (cid:17) + Y s =0 (cid:16) x + e πi ( k + s ) x + e πi ( k +2 s ) x (cid:17) + 3 Y n =0 (cid:16) i e iπn e iπ ( k + k ) x x x + e iπ ( k + k ) x x x (cid:17) (3.9)for some k , k , k , k ∈ Z / Z . We then find that the Fermat sextic fourfold contains thealgebraic cycles C k k k k nσ : x σ (0) + e πi k x σ (1) + e πi k x σ (2) = 0 x σ (3) + e πi k x σ (4) + e πi k x σ (5) = 0 i e iπn e iπ ( k + k ) x σ (0) x σ (1) x σ (2) + e iπ ( k + k ) x σ (3) x σ (4) x σ (5) = 0 , (3.10)9ith k i ∈ Z / Z and n ∈ Z / Z .There are further algebraic cycles of the form f = f = f = 0 contained in X whichcan be seen by constructing other factorizations of the form X : f P + f P + f P = 0 , (3.11)see [23] and [20] for a more systematic treatment. Note also that all of the examples ofalgebraic cycles we have given are complete intersections inside the ambient P , whichpoints to another direction of generalization: algebraic cycles which are not completeintersections. Over Q , it is known, however, that all of H , prim ( X ) is generated by theabove algebraic cycles [21, 23, 24]. Having introduced some algebraic cycles on the Fermat sextic, let us study some of theirproperties. We will limit our discussion mostly to the ‘linear’ algebraic cycles C ℓ σ .As each of the C ℓ σ is given by three linear equations inside P , each such cycle hasthe topology of P . To compute intersection numbers, we can use the following trick.Consider a complete intersection of X with x − αx = 0 for α = − and x − βx = 0 for α = β − . The resulting cycle on X is in the class H restricted to X . Using (3.4),however, we see that this cycle is reducible into a sum of six of the C ℓ σ . We may hencewrite H = X ℓ =0 C ℓ σ (3.12)for any choice of σ and every ℓ and ℓ . As H · H = 6 on X and H · C ℓ σ is the samefor every C ℓ σ by symmetry, it follows that H · C ℓ σ = 1 . (3.13)Using the observation that C ℓ σ · C ℓ ′ σ = 0 if ℓ and ℓ ′ differ in all three components (togetherwith a similar rule when intersection algebraic cycles employing different permutations σ ), the above can be iterated to find that the intersection numbers follow the patterndim ( C ℓ σ ∩ C ℓ ′ σ ′ ) C ℓ σ · C ℓ ′ σ ′ −
40 1 ∅ . (3.14)i.e. the dimension of the intersection of two algebraic cycles in P determines the inter-section number between the associated homology classes . For any pair of permutations If two algebraic cycles do not intersect transversely, there is always a pair of homologous (typicallynon-holomorphic) cycles that do intersect transversely. and σ ′ , the intersection numbers can also be expressed in terms of relations on the ℓ i and ℓ ′ j .Although one can work out the details using the same approach, such a simple patternis not obeyed by the other algebraic cycles introduced in the last section. Self-intersectionsof any algebraic cycle C f ,f ,f of complete intersection type given by f = f = f = 0 can however be worked out using adjunction, and the result is [20] C f ,f ,f · C f ,f ,f = d d d (36 − d + d + d ) + d d + d d + d d ) (3.15)where d i are the degrees of the polynomials f i . Let us now try to see how many moduli we expect to be stabilized by demanding thatany of the cycles C ℓ σ remains of type (2 , . We can work out the number of polynomialdeformations which are obstructed by demanding that C ℓ σ is an algebraic cycle as follows.First one observes that it does not matter which C ℓ σ we are talking about as they areall equivalent modulo automorphisms of the Fermat sextic. For α = − , let us henceconsider the cycle C : x − αx = 0 , x − αx = 0 , x − αx = 0 . (3.16)We can introduce a new set of coordinates: ( y , y , y , y , y , y ) = ( x − αx , x + αx , x − αx , x + αx , x − αx , x + αx ) , (3.17)in terms of which the Fermat sextic equation (3.4) becomes y y (3 y + 10 y y + 3 y ) + y y (3 y + 10 y y + 3 y ) + y y (3 y + 10 y y + 3 y ) = 0 (3.18)Polynomial deformations are counted by counting monomials of degree 6 modulo theJacobi ideal. There are (cid:0) (cid:1) = 462 possible monomials of degree 6 in 6 variables. TheJacobi ideal is generated by (3 y κ +1 +30 y κ +1 y κ +15 y κ +1 y κ , y κ +30 y κ y κ +1 +15 y κ y κ +1 ) with κ = 0 , , . (3.19)We can use the Jacobi ideal to eliminate all monomials proportional to y i , and there are such monomials. Hence the number of complex structure moduli is , which equals h , ( X ) as expected. Straightforwardly using C ℓ σ as a flux is at odds with primitivity and flux quantization. To achieve aprimitive flux, we would need to choose G = C ℓ σ − H . This is at odds with flux quantization, whichrequires G to be integral up to a piece H . One would hence need to consider G = 3 C ℓ σ − H . Anypiece proportional to H does however not influence complex structure deformations, and the number ofstabilized moduli is the same for C ℓσ and C ℓσ , so that we prefer to simply ask about the ‘Hodge Locus’of C ℓ σ here. This way of counting deformations is explained in some more detail in Section 4. It gives the sameresult as counting monomials modulo automorphism of P , but is more convenient here.
11e now demand the cycle C ℓ σ to persist as an algebraic cycle. This is the case only ifthe deformed fourfold is of the form y P ( y , ..., y ) + y Q ( y , ..., y ) + y R ( y , ..., y ) = 0 (3.20)where P, Q, R are homogeneous polynomial of degree . This means that we can useonly the monomials that have a factor of y , y or y to deform the Fermat sextic. Theobstructed deformations are then monomials of degree 6 in the three coordinates y , y , y .This gives (cid:0) (cid:1) = 28 deformations. We have to subtract the × monomials that arein the Jacobi ideal. We then obtain that moduli are fixed by demanding that any ofthe C ℓσ persists as an algebraic cycle.Again, there is a general version of this method that can be applied to any cycle C f ,f ,f . The result only depends on the degrees of the polynomials f i and can be foundin [20].The main issue with this approach which requires us to work harder is that we areinterested in stabilizing all complex structure moduli, which forces us to consider linearcombinations of algebraic cycles. Demanding that a single cycle C ℓ σ be algebraic only fixessome, but not all of the moduli. Similar results are obtained for other cycles C f ,f ,f , sothat we are led to consider linear combinations of (the Poincaré duals of) algebraic cycles.Merely counting polynomial deformations then becomes useless and we need a method toevaluate (2.11) in order to treat such situations.A further issue that deserves some discussion concerns the Hodge conjecture. Whilethe Hodge conjecture over Q has been proven for the Fermat sextic [23, 24] (see Section 4for more details), we do not know if it is true in general. This means for other points in themoduli space of the sextic or for other fourfolds, the number of polynomial deformationsthat are fixed by demanding a cycle stays algebraic may not equal the number of complexstructure deformations that are fixed by demanding that the dual integral (2 , formstays of type (2 , . Of course every algebraic cycle must be dual to an integral form oftype (2 , , but it is not clear that every integral form of type (2 , can be represented bya linear combination of algebraic cycles. In our context this implies that there might beextra flat directions that cannot be detected from polynomial deformations. Again, beingable to evaluate (2.11) settles this issue. In this section we will use the techniques of rational forms to explicitly describe the middlecohomology of X , some background on these techniques is given in Appendix A. This isthen used to describe complex structure deformations and moduli stabilization for fluxesdefined by (sums of) algebraic cycles. Throughout this section, X is the Fermat sexticfourfold (3.4). 12 .1 Middle cohomology from residues of rational forms As reviewed in Appendix A, primitive forms of Hodge type ( p, − p ) on the Fermat sexticare described as residues of rational forms ϕ = P ( x ) Q ( x ) − p Ω , (4.1)on P . Here Q = 0 is the hypersurface equation defining the Fermat sextic fourfold X , Ω is a fixed differential form on P that is completely antisymmetric in the homogeneouscoordinates x i , and P is a homogeneous polynomial of degree − p ) . The residue mapis linear, maps surjectively to the primitive forms in the middle cohomology of X , andbecomes injective when restricting to polynomials P which are not contained in the Jacobiideal of Q .Let us apply these statements to reproduce the Hodge numbers of the sextic. To workout the dimensions of the rings we are going to consider, it is beneficial to know that thereare degree l in m variables ) = (cid:18) l + m − l (cid:19) (4.2)terms in a homogeneous polynomial of degree l in m variables.The existence of a unique (4 , -form up to scaling follows from the fact that for p = 4 , P is just a number. To find classes of (3 , -forms, we hence need to consider the case p = 3 , i.e. homogeneous polynomials of degree modulo the Jacobi ideal of Q : H , prim ( X ) = C [ x , · · · , x ] h ∂ i Q i = C [ x , · · · , x ] h x , · · · , x i . (4.3)The ring of homogeneous polynomials of degree in variables has dimension . Atthe Fermat point, the Jacobi ideal is generated by the polynomials x i for all i , so that · generators of C [ x , · · · , x ] are contained in the Jacobi ideal of Q . We hencerecover the familiar number h , prim ( X ) = h , ( X ) = 426 . Finally, for H , prim ( X ) we have p = 2 , so that we need to count polynomials of degree modulo the Jacobi ideal: H , prim ( X ) = C [ x , · · · , x ] h x , · · · , x i . (4.4)We can work this out by noting that for each variable, the number of terms in C [ x , · · · , x ] which are in the ideal x i is given by a homogeneous polynomial of de-gree . Using this we need to take into account that for each pair of variables x i and x j there are terms x i x j P ( x ) for a polynomial P ( x ) of degree , which are in both the idealgenerated by x i and x j . Hence (cid:12)(cid:12)(cid:12)(cid:12) C [ x , · · · , x ] h x , · · · , x i (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) (cid:19) − · (cid:18) (cid:19) + 15 · (cid:18) (cid:19) = 1751 = h , prim ( X ) . (4.5)13 .2 Group actions and residues The content of the last subsection can be rephrased by considering the natural groupaction of G = µ /µ for µ = Z / Z by coordinatewise multiplication: ( x , · · · , x d +1 ) → ( ζ x , · · · , ζ d +1 x d +1 ) (4.6)for ζ ni = 1 an n -th root of unity. The quotient arises because elements of µ for which all ζ i are equal are inside the C ∗ acting on the homogeneous coordinates of P . Let us consider the character group A of G , which is the group of representations bycomplex valued functions of G : A = { a = ( a , · · · , a d +1 ) | a i ∈ Z / Z and X i a i = 0 mod 6 } . (4.7)The elements of A are functionals on G that associate to an element g ∈ G the phase a ( g ) = Y i ζ a i i . (4.8)Note that the condition P i a i = 0 mod guarantees that the unit element of G (i.e. ζ = ... = ζ d +1 ) is mapped to , i.e. that this is indeed a group homomorphism.The action of G on X induces an action on the middle cohomology H ( X, C ) . Onecan use the character group A to describe such an action. G is an abelian group, so itselements can be diagonalized simultaneously. The elements a play the role of ‘eigenvalues’.We may define ‘eigencycles’ relative to a ∈ A to be those classes η for which g ∗ η = a ( g ) η ∀ g ∈ G . (4.9)For a given a , we denote the span of the cycles which satisfy the above relation by V ( a ) .The spaces V ( a ) have the nice property that any pair V ( a ) and V ( a ′ ) is orthogonalexcept when a = − a ′ . To see this, take η a in V ( a ) and η a ′ in V ( a ′ ) . The inner form (givenby the integral of their wedge product) then transforms as Z X η a ∧ η a ′ → Z X η a ∧ η a ′ Y i ζ a i + a ′ i i ∀ g ∈ G . (4.10)However, as the inner form is merely a number which hence must be invariant under theaction of G , it follows that a = − a ′ is a necessary condition for the integral to be non-zero.To see the relation between the forms realized as residues and the eigenspaces underthe character group, let us use a monomial basis for the polynomials P in (4.1). For b = ( b , · · · , b ) , there is an associated monomial µ b = x b · · · x b with P i b i = deg P =6(4 − p ) . To such a monomial, we can associate a differential form ϕ a = µ b Q ( x ) − p Ω . (4.11) Note that this group is larger than the group µ which is used in the Green-Plesser mirror construc-tion: as we do not have a term proportional to Q i x i there is no need to impose Q i ζ i = 1 . a = (1 , , , , ,
1) + b . Under the group action (4.6), ϕ a has the simple transfor-mation behavior ϕ a → a ( g ) ϕ a (4.12)which follows from the fact that the Fermat polynomial Q of degree n is invariant and Ω transforms as Ω → Y i ζ i ! Ω . (4.13)As deg P = k · deg Q − k − , we hence have that | a | ≡ X i a i = 16 (6 + deg P ) = k = 5 − p . (4.14)We can furthermore associate µ b with a generator of C [ x , · · · , x ] | b | / h ∂ i Q i , as long as b i < . When this is satisfied we have a i < , so that we conclude that ϕ a ∈ H p, − p ( X ) . (4.15)This recovers the following result of [24–27], which can be phrased in the presentcontext as follows: Let A ∗ be the subset of the character group for which all of the a i = 0 .Then (Theorem 1 of [24]):a) dim C V ( a ) = 1 if and only if a ∈ A ∗ ; dim C V ( a ) = 0 otherwise.b) The Hodge type of V ( a ) is given by ( p, q ) = (5 − | a | , | a | − (4.16)and the canonical representative with ≤ a i ≤ should be chosen for each a i in theabove formula. Note that | a | is always an integer as P a i = 0 modulo . Togetherwith a), the above implies that for η a ∈ V ( a ) , ¯ η a is proportional to η − a .It is not hard to use this description to simply enumerate primitive forms by countingappropriate tuples a , one finds | a | elements in A ∗ (4.17) Elements of V ( a ) can not only be realized in terms of ϕ a but also by forming appropriatelinear combinations of algebraic cycles, which links the two descriptions of elements of15 , ( X ) . Furthermore, this allows us to work out the span of the algebraic cycles. Finally,finding representatives for all V ( a ) with | a | = 3 in terms of algebraic cycles proves theHodge conjecture (over Q ) for the Fermat sextic, see [21, 23, 24, 27] for more details andgeneralizations to other Fermat varieties.The description of forms η a ∈ V ( a ) in terms of algebraic cycles works by puttingrestrictions on the tuples a ∈ A ∗ . We call an element a ∈ A ∗ n -decomposable if theelements of a can be decomposed into pairs such that maximally n of them satisfy a i + a j = 0 (4.18)modulo . For the Fermat sextic, a ∈ A ∗ with | a | = 3 , so that it corresponds to a (2 , -form, can be -decomposable, -decomposable, or indecomposable . Using a computermakes it easy to enumerate them, the resulting numbers and their general forms (up topermutations and taking the inverse) are given belowtype number standard form − decomposable r, − r, s, − s, t, − t )1 − decomposable
720 ( t, − t, , , , indecomposable
30 (1 , , , , , (4.19)As they should, these sum up to the total primitive classes in H , ( X ) .Let us consider -decomposable elements of A ∗ . We can write a general 3-decomposable a as a σ (0) + a σ (1) = 0 a σ (2) + a σ (3) = 0 a σ (4) + a σ (5) = 0 , (4.20)for some permutation σ . The corresponding element of V ( a ) is η a = X ℓ ℓ ℓ e − iπ ( a σ (1) ℓ + a σ (3) ℓ + a σ (5) ℓ ) C ℓ σ , (4.21)where C ℓ σ are the linear algebraic cycles defined in (3.5). Using the transformation behaviorof the C ℓ σ it is not hard to see that it is crucial for the defining relation (4.9) of eigencyclesto hold that we are only talking about 3-decomposable a here.This result can be immediately used to constrain the possible intersections betweenthe C ℓ σ and the residues of the forms ϕ a , and we shall see how these are in fact fixed up tonormalization later. A second application concerns the linear relations between the C ℓ σ .We have already seen that they obey the ‘sum rule’ (3.12) using elementary methods. Thisis insufficient to work out the dimensionality of span of all of the C ℓ σ , however. The aboveproves that its dimension is 1001 and shows how further linear relations arise: whenever a is 3-decomposable in more than one way, we can write down η a in two independent waysin terms of the C ℓ σ using different permutations. As V ( a ) is complex one-dimensional, thisimplies that the two expressions must be proportional.Following the formulae in [23], it is possible to write down similar expressions foreigencycles for 1- or in-decomposable a using the non-linear algebraic cycles (3.8) and(3.10). As P a i = 0 mod , -decomposable implies -decomposable. .4 Complex structure moduli Having explained how to capture the middle cohomology in terms of residues and sketchedthe relationship to algebraic cycles, let us now discuss complex structure deformationsin this language. We focus again on the Fermat sextic hypersurface in P and considerdeforming away from the Fermat locus. We may parametrize a general deformation as Q ( x ; s ) = X i x i + X b I s I µ b I (4.22)for complex parameters s I and monomials µ b I = x ( b I ) · · · x ( b I ) (4.23)which are such that | b I | = 1 and ( b I ) i < .This corresponds to complex structure deformations, which may be represented bydeformations of the holomorpic top-form Ω , which in turn can be written as a residue Ω( s ) = Res (cid:20) Ω Q ( x ; s ) (cid:21) = Res [ ϕ ] (4.24)throughout the moduli space. Setting s = 0 in the above, we recover the holomorphictop-form at the Fermat locus.The variation of Hodge structure is described by choosing a topological basis γ k of H ( X ) and studying the variation of the integrals Z γ k ϕ = Z γ k Res (cid:20) PQ ( x ; s ) − p Ω (cid:21) . (4.25)as we vary Q . This defines the Hodge bundle and we may locally choose a trivializationby identifying the topological cycles γ k in nearby sextics. There is a flat connection ∇ I on this bundle, called the Gauss-Manin connection, which acts on residues as ∇ I ϕ = Res (cid:20) ∂ I PQ ( x ; s ) − p Ω (cid:21) . (4.26)The flatness of this connection simply follows from the commutativity of the differentialoperators.An infinitesimal deformation of Ω =
Res [ ϕ ] = Res (cid:20) Q ( x ; s ) Ω (cid:21) (4.27)at the Fermat point can hence be written as ϕ = ϕ | s =0 + X I s I ∂ s I ϕ | s =0 . (4.28)17ote that ∂ I ϕ | s =0 = − µ b I Q ( x ; 0) Ω = − ϕ a I , (4.29)which gives a (3 , -form upon taking the residue at the Fermat point as we have restrictedto b i < in (4.22). We hence recover that deformations of the complex structure are givenby (3 , -forms.One could also include terms in the sum in (4.22) for which µ b I is in the Jacobi idealof Q . Deforming by such terms again adds a term to Ω which is given by the residue ofa rational form, but now the pole order of this rational form can be reduced to (seeAppendix A). This implies that the residue does not produce a (3 , form, but a (4 , form. Such deformations would hence only rescale Ω .In physics, one is usually interested in the covariant derivative D I = ∇ I + ∂ I K , whichby definition maps D : H p, − p → H p − , − p +1 . (4.30)When working at the Fermat point and acting on H , , we have just seen that we musthave ∂ I K = 0 | s =0 as ∇ I alone already has the property of mapping purely to H , whenusing the basis of monomials b I with ( b I ) i < to define local coordinates on the complexstructure moduli space . In the monomial basis we have chosen, the action of covariantderivatives is hence particularly simple.This structure becomes slightly more complicated when considering second derivativesof Ω ∂ I ∂ J ϕ ( s ) | s =0 = 2 µ b I + b J Q Ω (cid:12)(cid:12)(cid:12)(cid:12) s =0 = 2 ϕ + b I + b J | s =0 . (4.31)As long as all components of b I + b J are smaller than , this form is of pure type (2 , .Whenever this is not the case, however, µ b I + b J is in the Jacobi ideal of Q and we mayreduce the pole order leading to a form of degree (3 , . To define a covariant derivativeacting on (3 , -forms, we need to subtract the (3 , -pieces of the derivatives. This meanswe need to set the derivative to zero whenever it produces a form for which µ b I + b J is inthe Jacobi ideal of Q .In summary, the covariant derivative acts on forms as D I : Res [ ϕ a ] → Res [ ϕ a + b I ] (4.32)as long as ( a + b I ) i < for all i and it sends them to zero otherwise.We need to evaluate the rank of the matrix (2.11) in order to find the (co)-dimensionof the Hodge locus and hence the number of stabilized moduli. From the above it followsthat it can be simply written as G IJ = D I D J Z X G ∧ Ω = 2 Z X G ∧ Res (cid:20) µ b I + b J Q Ω (cid:21) (4.33) Note that this does not imply that ∂ I K = 0 holds for any choice of coordinates on complex structuremoduli space, as such coordinate changes can give Kähler transformation that map K ( s, ¯ s ) → K ( s, ¯ s ) + f ( s ) + ¯ f (¯ s ) for a holomorphic function f ( s ) . (2 , piece of the derivatives acting on Ω . Ina similar vein, any term for which one of the ( b I + b J ) i ≥ vanishes. See [19, 20, 28–30]for an in-depth discussion of the above result. In order to evaluate the integral in (4.33), we need to know the period integrals of algebraiccycles, i.e. for an algebraic cycle C , we need to know Z C Res (cid:20) µ b I + b J Q Ω (cid:21) . (4.34)The periods of forms such as (4.31) over the linear algebraic cycles C ℓ σ have beencomputed in [12, 13] using results of [28]. The upshot is that for | b | = 2 , we have that πi ) Z C ℓ σ µ b Q Ω = sgn( σ )6 e iπ ( P e =0 ( b σ (2 e ) +1)(2 ℓ e +1) ) if b σ (2 e − + b σ (2 e − = 40 otherwise. (4.35)Up to the overall normalization, this can also be derived by using the automorphism group ( Z / Z ) / ( Z / Z ) ⋊ S of the sextic. One must have that σ ◦ g (cid:18)Z C ℓ σ µ b Q Ω (cid:19) = Z C ℓ σ µ b Q Ω g ∈ ( Z / Z ) / ( Z / Z ) , σ ∈ S , (4.36)with σ ◦ g (cid:16)R C ℓ σ µ b Q Ω (cid:17) ≡ R ( C ℓ σ ) ′ µ ′ b Q Ω ′ , where the prime quantities are the ones transformedby g and σ .Let us first consider permutations. After acting with any permutation σ , we maysimply relabel the coordinates x i in the rhs of (4.36) to undo the permutation again.This produces the same expression we started from, except for Ω , which produces a sign sgn( σ ) as it is completely antisymmetric in the x i . This explains the corresponding factorin (4.35).Now consider the action by g ∈ G = ( Z / Z ) / ( Z / Z ) . This will both act on thedifferential form under the integral, as well as the cycle C ℓ σ . We can write g (cid:18)Z C ℓ σ µ b Q Ω (cid:19) = Z ( C ℓ σ ) ′ µ ′ b Q Ω = a ( g ) Z C ℓ ′ σ µ b Q Ω (4.37)for some ℓ ′ , where a is the element of the character group associated with a = b +(1 , , , , , . To check if this makes (4.36) consistent with (4.35), it is enough to con-sider one of the generators of G , all other cases can be found by analogous computationsor repeated application of this action. As we have already understood the action of per-mutations, let us furthermore choose σ as the trivial permutation σ = id and investigate19ntegrals over the cycles C δ id . Consider the map ζ : x → e iπ/ x , which generates one ofthe ( Z / Z ) ⊂ G . ζ maps C ℓ σ to C ℓ ′ σ where δ ′ = δ − . We can hence write a ( ζ ) Z C δ ′ id µ b Q Ω = e iπ a Z C δ ′ id µ b Q Ω = e iπ ( b +1) e iπ [ ( b +1)(2 ℓ ′ +1)+( b +1)(2 ℓ ′ +1)+( b +1)(2 ℓ ′ +1) ]= e iπ ( b +1) e iπ [( b +1)(2 ℓ − b +1)(2 ℓ +1)+( b +1)(2 ℓ +1)] = 16 e iπ [( b +1)(2 ℓ +1)+( b +1)(2 ℓ +1)+( b +1)(2 ℓ +1)] = Z C δ id µ b Q Ω (4.38)that is exactly what (4.36) says.We then need to know the integral of ϕ a on a single cycle C ℓ σ to compute the integralsof ϕ a over all the cycles C ℓ ′ σ in the same orbit. One simply uses Z C ℓ ′ σ ϕ a = a ( g ) − Z C ℓ σ ϕ a . (4.39)that is derived by (4.36). This shows that in fact all relative coefficients of (4.35) are fixedby G ⋊ S , as it acts transitively on the C ℓ σ . It is not true, however, that G ⋊ S actstransitively on a basis of algebraic cycles for H , ( X ) ∩ H ( X, Q ) . If we want to studyperiods of such a basis up to a global normalization, we hence need more than the relativefactors between periods of the C ℓ σ .Note that the condition b σ (2 e − + b σ (2 e − = 4 for all e ∈ { , , } implies that theintersections of C ℓ σ with the eigenspace V ( a ) is non-zero only if a is 3-decomposable. Thisis not unexpected, as we have seen, V ( a ) for a C ℓ σ , whereas eigenspaces for a V ( a ) and V ( a ′ ) are orthogonal except a ′ = ¯ a . This can also be seen directly asfollows. Let us consider the case where σ is the trivial permutation. The action of ζ k ζ k on C ℓσ is trivial in this case. It follows that Z C ℓ σ µ b Q Ω = ζ k ζ k Z C ℓ σ µ b Q Ω = e iπ · k ( b + b +2) Z C ℓ σ µ b Q Ω (4.40)so that the integral can only be non-zero when a + a = 0 mod 6 . We can make thesame argument for the other two pairs x , x and x , x . The same argument applies(with different pairings) for other permutations, and implies that a = b + 1 must be3-decomposable for the integral to be non-zero.The above can be generalized to arbitrary algebraic cycles of complete intersectiontype [13], i.e. cycles of the type f = f = f = 0 inside a hypersurface (3.11). The result20s πi ) Z Z µ b Q Ω = c · , (4.41)where c is the unique number which satisfies µ b det( ∂ i H j ) = c det (Hess( Q )) mod h ∂ i Q i , (4.42)the vector H is given by H = ( f , g , f , g , f , g ) and Hess denotes the Hessian matrix.For the linear cycles C ℓσ , this reproduces the normalization of (4.35) from the generalformula (4.41). With the tools we have collected in the previous section, we are now ready to directlyaddress how algebraic cycles can be used as fluxes and how many moduli they stabilize.All one needs to do after defining a flux which is appropriately quantized and primitive,is to evaluate the period integrals (4.34) needed to compute the rank of the × matrix G IJ (4.33). As a first example, let us revisit the case of using a single linear algebraic cycle C ℓ σ (3.5)as a G flux. Using the period integrals (4.35) we findrk G IJ (cid:0) C ℓ σ (cid:1) = 19 , (5.1)which is precisely the same number we obtained by analyzing obstructed polynomialdeformations.In fact, this is the lowest possible value G IJ can have for any algebraic cycle. This isnot surprising as linear algebraic cycles are the simplest type that can exist for the Fermatsextic (see Proposition 7 ‘Olympiad problem’ of [30]). With the material we have collected, it is straightforward to work out what happenswhen we add two different linear algebraic cycles C ℓ σ + C ℓ ′ σ ′ . A direct computation (incombination with the automorphism group) shows that the rank of G IJ only depends on The same issues as discussed in footnote 8 apply. C ℓ σ · C ℓ ′ σ ′ rk G IJ (cid:0) C ℓ σ + C ℓ ′ σ ′ (cid:1)
21 19 − . (5.2)The first row corresponds to the case C ℓ σ = C ℓ ′ σ ′ . As the number of flat directions for alinear combination of two cycles is at least equal to the number of flat directions commonto both of them, the rank of the matrix G IJ must be subadditive:rk G IJ (cid:0) C ℓ σ (cid:1) + rk G IJ (cid:16) C ℓ ′ σ ′ (cid:17) ≥ rk G IJ (cid:16) C ℓ σ + C ℓ ′ σ ′ (cid:17) , (5.3)which is indeed the case for the numbers we find. The sextic moduli space has the symmetry group G = µ /µ for µ = Z / Z , that wediscussed in Section 4.2. Consider the Greene-Plesser subgroup G P G = ( Z / Z ) [31]. It isgenerated by α i = 1 for i = 1 , , , with action ( x , x , x , x , x , x ) → (( α α α α ) − x , α x , α x , α x , α x , x ) (5.4)on the homogeneous coordinates of X . Famously, only a single complex structure de-formation, corresponding to the monomial Q i x i , i.e. b = (1 , , , , , , is symmetricunder the action of this group, while all others are projected out. The obvious way toconstruct a flux that is even under the action of G P G is to start with the orbit of any ofthe linear cycles C ℓ σ under G P G . It turns out that this is not the minimal choice and onecan repeatedly use the sum rule (3.12) to show that X g ∈ G GP g ( C ℓ σ ) = 4 C eee (5.5)where C eee = X ℓ ,ℓ ,ℓ ∈ (0 , , C ℓ σ . (5.6)Using (3.12) and the intersection numbers (3.14) one can also show directly that C eee iseven under G P G and that C eee · C eee = 3 . As there are terms in the sum, we have C eee · H = 27 . A symmetric flux that is primitive and properly quantized is G sym4 = C eee − / H , (5.7)and the induced tadpole is hence / , which is well within the allowed range.22irectly evaluating G IJ ( C eee ) using (4.35) we find that indeedrk G IJ ( G sym4 ) = 141 . (5.8)In particular, there is a single entry that obstructs the unique deformation that is sym-metric under G P G . Hence the symmetric flux fixes the invariant modulus; this correspondsto solving D s i W = 0 explicitly (see at the end of Section 2 for notation) and finding thatthe solution sits at the Fermat point. Our computation goes further: it is true that theFermat point (that belongs to the fixed point set of G P G ) is a solution of D s ni a W = 0 (with a = 1 , ..., ), but only out of the non-symmetric deformations under G P G arefixed by G sym4 , the other ones are flat directions.Finally, one may wonder about using a flux that is symmetric under the entire auto-morphism group ( Z / Z ) / ( Z / Z ) ⋊ S of the sextic at the Fermat point. As all of theforms ϕ a with | a | = 3 have non-trivial transformations already under the scaling part,it follows that the matrix G IJ can only contain zeros in such a case. The same can beseen by noting that ( Z / Z ) / ( Z / Z ) ⋊ S acts transitively on the C ℓ σ . Using the sum rule(3.12) one can argue that summing over an orbit results in a cycle proportional to H (the only invariant cycle), so that G IJ vanishes for all I, J . This implies that there are noinvariant fluxes that are primitive for this group.
Let us now see if we can find a flux stabilizing all moduli employing only linear algebraiccycles. From the subadditivity (5.3), it follows that we need to consider a linear combi-nation of at least of the C ℓ σ . As we have seen in Sections 4.2 and 4.3, only a subspaceof dimension within H , prim ( X ) is spanned by the linear algebraic cycles, and thissubspace precisely corresponds to 3-decomposable tuples a . By a simple scan over allpossibilities, one can find that for every b I with | b I | = 1 , there is a b J with | b J | = 1 suchthat a IJ = (1 , , , , ,
1) + b I + b J (5.9)is 3-decomposable. In other words, linear cycles in principle allow us to constrain allcomplex structure deformations.Due to the large number of linear algebraic cycles, there are ·
15 = 3240 of them, asimple scan is computationally much too expensive. Besides randomly sampling choices,the following (semi-)systematic method can be used. It is an experimental fact that theinequality (5.3) becomes an equality if we consider sums of linear algebraic cycles suchthat all of them are mutually orthogonal, i.e.rk G IJ X i ∈ I C ℓ i σ i ! = | I | · if C ℓ i σ i · C ℓ j σ j = 0 (5.10)for all i = j ∈ I . Correspondingly, the maximal size of a set with this property is , one23ossible choice being I max = { [[0 , , , [0 , , , , , , [[0 , , , [0 , , , , , , [[0 , , , [0 , , , , , , [[0 , , , [0 , , , , , , [[0 , , , [0 , , , , , , [[1 , , , [0 , , , , , , [[1 , , , [0 , , , , , , [[1 , , , [0 , , , , , , [[1 , , , [0 , , , , , , [[1 , , , [0 , , , , , , [[2 , , , [0 , , , , , , [[2 , , , [0 , , , , , , [[2 , , , [0 , , , , , , [[2 , , , [0 , , , , , , [[3 , , , [0 , , , , , , [[3 , , , [0 , , , , , , [[3 , , , [0 , , , , , , [[3 , , , [0 , , , , , , [[3 , , , [0 , , , , , , [[4 , , , [0 , , , , , , [[5 , , , [0 , , , , , , [[5 , , , [0 , , , , , } , (5.11)where each entry is of the form [ ℓ i , σ i ] . Consistent with (5.10), a sum of the associatedlinear algebraic cycles gives a matrix G IJ of rank .We can use this as a starting point to construct a flux stabilizing all moduli by addinga further linear algebraic cycle. We are looking for a C e such that for C = X i ∈ I max C ℓ i σ i + C e (5.12)the flux G = C + nH (5.13)is primitive and appropriately quantized. Quantization requires n to be half-integer, n = m/ for m odd, and primitivity requires m = 22 + C e · H , (5.14)so that we can choose C e = − C ℓ e σ e , which gives m = 7 . The tadpole of such a configurationis given by N D = X i ∈ I max C ℓ i σ i − C ℓ e σ e − H ! = X i ∈ I max C ℓ i σ i − C ℓ e σ e ! − (5.15)which is minimal if we choose C ℓ e σ e · P i ∈ I max C ℓ i σ i to be as large as possible. There is aunique such C ℓ e σ e which has C ℓ e σ e · P i ∈ I max C ℓ i σ i = 11 , it is given by [ σ e , ℓ e ] = [[2 , , , [0 , , , , , . (5.16)Working out G IJ ( C ) = G IJ ( G ) one finds that it has maximal rank, . The resultingtadpole is computed to be N D = G ∧ G = 7754 . (5.17)Although such a flux would stabilize all complex structure moduli at the Fermat point, itsignificantly overshoots the available tadpole χ ( X ) /
24 = 435 / . (5.18)24 .5 Tadpole issues The result of the last section is at the same time encouraging and disappointing: while itis not hard to find a primitive flux with proper quantization that stabilizes all moduli, itgenerates a tadpole that is almost twice the maximal allowed value. Scanning over randomlinear combinations of linear algebraic cycles gives many more examples with the sameproperties. This result can already be anticipated from rough estimates.Consider stabilizing all moduli by a combination of linear algebraic fluxes C = X i ∈ I f i C ℓ i σ i (5.19)and let P f i = 3 m for an odd integer m . We can then find a primitive properly quantizedflux by setting G = C − m H . (5.20)The induced tadpole is then G = (cid:18) C − m (cid:19) . (5.21)Ignoring the contribution from C ℓ i σ i · C ℓ j σ j to C , and assuming that all f i = 1 , we can writethis as G = (cid:18) m − m (cid:19) . (5.22)As every C ℓ σ stabilizes at most moduli, we need to have m > / . This roughlyreproduces (5.17) for the minimal choice of m .The negative contribution in (5.22) points at a potential way out by letting m becomesufficiently large. As we have seen, there are at most mutually orthogonal linear al-gebraic cycles and we need to let m be significantly larger to bring the tadpole downsufficiently. It turns out that ignoring mutual intersections between the terms in C becomeincreasingly unjustified, so that the tadpole contribution of such fluxes is again far toolarge to give a viable model.Until now, we have completely ignored non-linear algebraic cycles. Performing a similarrough estimate gives a comparable result to what we have found for the linear algebraiccycles. There, the crucial ratio was that of the number of moduli that could be fixed witha single linear algebraic cycle , to the square of such a cycle, . These ratios can alsobe computed for non-linear algebraic cycles, the result is that for a complete intersectionalgebraic cycle C f f f given by f = f = f = 0 for homogeneous polynomials with25egrees d i [32] ( d , d , d ) C f f f rk G IJ ( C f f f )(1 , ,
1) 21 19(1 , ,
2) 34 32(1 , ,
3) 39 37(1 , ,
2) 56 54(1 , ,
3) 66 62(1 , ,
3) 81 71(2 , ,
2) 96 92(2 , ,
3) 120 106(2 , ,
3) 162 122(3 , ,
3) 243 141 . (5.23)As the ratio of these two numbes stays roughly the same, we can anticipate to find similarresults using non-linear algebraic cycles. As the maximal allowed tadpole of the flux is χ ( X ) /
24 = 435 / C , but we need to stabilize moduli, the ratio between C andrk G IJ ( C ) should be roughly rather than the ratio of ∼ (and larger) observed above.The above results do not imply that there cannot be a properly quantized and primitiveflux stabilizing all moduli that also satisfies the tadpole constraint. Much more work isneeded to make such a claim. What we can say (at least for the sextic fourfold we studied),however, is that it is not completely straightforward to construct such a flux. In this section we address the problem of moduli stabilization in cases with symmetry insome more generality. In particular, we explain why fluxes that are invariant under somegroup actions typically leave some flat directions in the effective potential. This will givea conceptual way of understanding the result found in Section 5.3.Let X p be a Calabi-Yau fourfold at a point p in its complex structure moduli space,and G any subgroup of the automorphism group of X p . Although what we are going tosay can be put in slightly more general terms, let us assume for simplicity that X p isa hypersurface in a toric variety T for which we can represent all forms in the middlecohomology of X p as residues. Let us furthermore assume that G acts by rescaling thehomogeneous coordinates x i of the ambient space by roots of unity and preserves theholomorphic top form Ω X | p at p (as well as the Kähler form of X p ).We can write a family in the vicinity of p as X : Q = Q + X N s N µ N + X Φ t Φ ν Φ = 0 (6.1)where X p is given by s = t = 0 , the monomials µ N are invariant under the action of G ,and the monomials ν Φ are not, but transform as ν Φ → α Φ ( g ) ν Φ (no summation) . (6.2)26y assumption we can write Ω X | p = Res (cid:20) Q Ω T (cid:21) , (6.3)for some fixed holomorphic form Ω T on the ambient space T . As both Ω X at p and Q are invariant under the action of G , it follows that Ω T must be preserved by G as well.Let us now consider switching on a flux G which is invariant under the action of G .This implies that the GVW superpotential (1.1) is invariant under G as well. The F-termsin the non-invariant directions t Φ at p transform as F Φ = Z X p G ∧ D Φ Ω X | p = − Z X p G ∧ Res (cid:20) ν Φ Q Ω T (cid:21) → α Φ ( g ) F Φ . (6.4)However, the above integral at p simply yields a number that cannot change under anyautomorphism, so that it follows that F Φ = 0 for all Φ . This argument was used in [33–36]to argue that the F-term equations in the non-invariant directions are automaticallysatisfied and one only needs to take care of the invariant directions.As we have seen in the example in Section 5.3, this does not imply that there are noflat directions along the non-invariant directions t Φ in complex structure moduli space. Wecan repeat a similar argument as above for the matrix G IJ to see why. Let us first considerthe mixed terms G N Φ between invariant and non-invariant directions. They transform as G N Φ = Z X p G ∧ Res (cid:20) µ n ν j Q Ω T (cid:21) → α Φ ( g ) G N Φ . (6.5)As these have a non-trivial scaling they must vanish, so that G IJ is block-diagonal amonginvariant and non-invariant directions. For the matrix elements between two non-invariantdirections we find G ΦΞ = Z X p G ∧ Res (cid:20) ν Φ ν Ξ Q Ω T (cid:21) → α Φ ( g ) α Ξ ( g ) G ΦΞ . (6.6)These can hence only be non-vanishing if α Φ ( g ) α Ξ ( g ) = 1 for all g ∈ G . This is a strongcondition, and for many ν Φ there is no ν Ξ such that it holds, which implies that both t Φ and t Ξ are flat directions.Let us now come back to the example discussed in Section (5.3), where X p is the sexticfourfold at the Fermat point and G is the Greene-Plesser group G GP . In this case thereis only a single invariant monomial µ N = Q i x i . All other monomials ν Φ correspond tonon-invariant directions. In order to have a non-zero matrix elements G ΦΞ we need that ν Φ ν Ξ = Y i x i , (6.7) These papers deal with Calabi-Yau threefolds. However their arguments and their conclusions directlyapply to fourfolds as well.
27s this is the only monomial of appropriate degree that is invariant under G GP . The onlynon-invariant complex structure deformations that have non-zero elements in G IJ hencecorrespond to pairs of tuples b Φ and b Ξ different from (1 , , , , , with P i ( b Φ ) i = 6 and P i ( b Ξ ) i = 6 such that b Φ + b Ξ = (2 , , , , , . (6.8)It turns out that there are precisely such pairs, so that the rank of G IJ can be at most for any flux that is symmetric under G GP , i.e. there are at least flat directions inthis case. For the example we have chosen in Section 5.3, this is precisely what was foundby explicitly evaluating G IJ . In this work we have begun to explore how to use algebraic cycles as fluxes on Calabi-Yaufourfolds. We have reviewed methods which allow us to compute the number of stabilizedmoduli and the induced tadpole for fluxes proportional to linear combinations of algebraiccycles. We have analyzed in detail the sextic fourfold. We have found fluxes that stabilizeall the complex structure moduli at a specific point in the complex structure modulispace, without the need of dealing with Picard-Fuchs equations of (very) high rank. Whatis striking about this analysis is that it appears very hard to find a flux that satisfies allconsistency constraints and stabilizes all of the complex structure moduli. In particular,in the example we have considered, we have noticed tension between tadpole cancellationand the desire to stabilize all complex structure moduli.The above is far from a complete analysis, and there are several crucial points that needto be addressed for a complete picture. First of all, it is in principle straightforward (buttedious) to work out rk G IJ ( C Σ ) , C , and H · C Σ for any linear combination of algebraiccycles C Σ . Having access to all fluxes defined via algebraic cycles is not sufficient, asthe integral Hodge conjecture for the Fermat sextic is presently unanswered. It has beenshown to be correct, however, for the quartic and quintic Fermat fourfolds in [37], andit is possible to extend their methods to the sextic. With a proof of the integral Hodgeconjecture for the Fermat sextic, it is then possible to compute rk G IJ ( G ) for all fluxessatisfying the tadpole constraint. This naively seems like a task that is computationallytoo demanding to be undertaken, but a clever exploitation of the large automorphismgroup of the sextic might make it feasible. We intend to attack this problem in futurework.Thinking even further ahead, it is highly desirable to extend the methods we have re-viewed to other points in the moduli space of the sextic with maximal H , ( Z ) ∩ H ( X, Z ) ,and even to other Calabi-Yau fourfolds. In particular, it would be exciting to find criteriawhich can distinguish which points in the moduli space can and which cannot be stabi-lized using fluxes that satisfy the tadpole constraint. Such criteria would have far reachingimplications for the existence and structure of the string landscape.28inally, let us note that the Fermat sextic we considered in this paper is known tobe modular [38]. In recent work, it was conjectured that in fact all flux vacua correspondto modular varieties [39], although the converse to this statement is not true [40]. Asimilar statement holds for attractive K surfaces [41], which appear in the study of fluxvacua on the ’toy’ fourfold K × K [10] (see also [8]), as well as the closely relatedattractor points on Calabi-Yau manifolds [42]. At its core, modularity is concerned withGalois representations, which are again related to algebraic cycles according to the Tateconjecture. It should be fascinating to explore this relationship further. Acknowledgements
We wish to thank Hossein Movasati for discussion and explanations. The work of R.V. ispartially supported by “Fondo per la Ricerca di Ateneo - FRA 2018” (UniTS) and byINFN Iniziativa Specifica ST&FI.
A Rational Forms, Residues, and Cohomology of Hy-persurfaces
In this section we review some classic material concerning rational differential forms on P n , i.e. forms with poles, and their residues. This is based on [43,44], a beautiful expositionof which can be found in [20, 45–47].The basic idea of residues of forms is to extend the residue formula πi Z γ dzz = 1 , (A.1)for γ a closed curve encircling the origin, to integrals of differential forms, i.e. πi Z γ dz ∧ αz = α , (A.2)for a smooth differential form α . In this way, rational differential forms on C n with polesalong z = 0 are naturally identified with smooth forms on the locus z = 0 . The followingessentially deals with properly formulating this idea for hypersurfaces X of P d +1 . Theupshot is that we can write differential forms on X as differential forms with poles on P d +1 .The setting we will be interested concerns differential forms with poles (‘rationalforms’) on complex projective space. By a result of [43], rational d + 1 -forms on P d +1 can always be written in terms of the unique holomorphic d + 1 -form Ω ϕ = P ( x ) R ( x ) Ω (A.3)29or homogeneous polynomials P ( x ) and R ( x ) with deg R = deg P + ( d + 2) . The form Ω is given by Ω = d +1 X j =0 ( − j x j dx ∧ · · · ∧ c dx j ∧ · · · ∧ dx d +1 , (A.4)where our notation is supposed to indicate that c dx j is omitted from the ∧ product of dx j .There are similar expression for n -forms with n < d + 1 . The constraint on the degrees of P and R hence guarantees that ϕ is invariant under the C ∗ acting on the homogeneouscoordinates x i .Depending on the choice of the denominator, ϕ can have poles of various orders alonga hypersurface X ⊂ P d +1 . Working modulo exact forms, the pole order can sometimes bereduced, as summarized in the following statements [43, 44].a) For any rational d + 1 -form ϕ there exists a η such that ϕ + dη has pole order d + 1 .b) If a rational d + 1 -form ϕ has pole order k along X and there exists and η such that ϕ + dη has pole order k − , we can choose η to have pole order k − .c) If X is given by Q ( x ) = 0 and ϕ has pole order k we can write ϕ = P ( x ) Q ( x ) k Ω . (A.5)There exists an η such that ϕ + dη has pole order k − if and only if P ( x ) is containedin the Jacobi ideal of Q ( x ) , i.e. the ideal generated by the polynomials ∂Q/∂x j .Elements of H d +1 ( P d +1 − X ) can hence be represented by forms such as (A.5). If X is described by a polynomial Q ( x ) = 0 and Q ( x ) has degree l , then deg P = kl − ( d + 2) .As we always reduce the pole degree of ϕ modulo exact forms, k is at most d + 1 , due tothe property a).The residue map is defined asRes : H d +1 ( P d +1 − X ) → H d prim ( X ) (A.6)as follows. For any d -cycle Γ on the hypersurface X we set Z Γ Res ( ϕ ) = Z T (Γ) ϕ , (A.7)where T (Γ) is a tube, i.e. a circle bundle over Γ . It can be shown that such a tube alwaysexists and the definition of the residue is independent of this choice. The image of theresidue map is not all of H d ( X ) , but only maps to the primitive cohomology H d prim ( X ) , i.e.to those forms perpendicular to the restriction of the hyperplane class. As shown in [43]the residue map is surjective on the primitive cohomologyim ( Res ) = H d prim ( X ) . (A.8)30et us see in some more detail how this definition of the residue realizes (A.2). Considera rational n -form of pole order k in a small neighborhood containing Q = 0 . There wecan choose coordinates such that ϕ becomes ϕ = dQ ∧ αQ k + βQ k − = 1 k − d (cid:18) αQ k − (cid:19) + β + k − dαQ k − (A.9)for some smooth forms α and β . Hence we may always reduce the pole order of holomorphicforms locally. Using a partition of unity, one can show that this can in fact be done globally,but at the expense of holomorphicity. Iterating this procedure, it follows that we may write(up to exact forms) ϕ = γ ∧ dfQ + δ (A.10)for some smooth forms γ and δ . The residue is then simplyRes ( ϕ ) = γ | X . (A.11)This explains why the residue of a holomorphic rational form ϕ such as (A.5) on P n isnot necessarily holomorphic, except when k = 1 .Let us now define A d +1 k to be the additive group of rational d + 1 -forms of pole orderat most k along X . We can then form the ‘cohomology groups’ H k ( X ) = A d +1 k ( X ) dA dk − ( X ) . (A.12)The H k ( X ) for different k have a filtration H ⊂ H ⊂ · · · ⊂ H d +1 , (A.13)which precisely maps to the Hodge filtration of the primitive cohomology under the residuemap: Res ( H k ) = F d +1 − k H d prim ( X ) , (A.14)where F d +1 − k H d = M i ≥ d +1 − k H i,d − i ( X ) . (A.15)Hence the residue map takes H ( X ) to H d, ( X ) , while H ( X ) maps to H d, ( X ) ⊕ H d − , ( X ) , etc. The forms of maximal pole order, k = d + 1 , are mapped to F H d prim ( X ) = H d prim ( X ) .We can isolate the Hodge cohomology groups of primitive forms by forming the quo-tients H p,d − p prim ( X ) = F p H d prim ( X ) F p +1 H d prim ( X ) = H d +1 − p ( X ) H d − p ( X ) . (A.16)where the last equality is realized by applying the residue map.31he result of the above is that we can associate Hodge cohomology groups in themiddle cohomology with polynomials P of appropriate degree modulo the Jacobi idealof the polynomial Q defining the hypersurface equation. 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