CERNTH/96269 hepth/9610162
Mirror Symmetry, Superpotentials and
Tensionless Strings on Calabi–Yau FourFolds
P. Mayr
Theory Division, CERN, 1211 Geneva 23, Switzerland
We study aspects of Calabi–Yau fourfolds as compactification manifolds of Ftheory, using mirror symmetry of toric hypersurfaces. Correlation functions of the topological field theory are determined directly in terms of a natural ring structure of divisors and the period integrals, and subsequently used to extract invariants of moduli spaces of rational curves subject to certain conditions. We then turn to the discussion of physical properties of the spacetime theories, for a number of examples which are dual to heterotic theories. Noncritical strings of various kinds, with low tension for special values of the moduli, lead to interesting physical effects. We give a complete classification of those divisors in toric manifolds that contribute to the nonperturbative fourdimensional superpotential; the physical singularities associated to it are related to the apppearance of tensionless strings. In some cases nonperturbative effects generate an everywhere nonzero quantum tension leading to a combination of a conventional field theory with light strings hiding at a low energy scale related to supersymmetry breaking.
October 1996 CERNTH/96269
1. Introduction
Dualities between perturbatively different string theories in various dimensions have led to a considerable improvement of the understanding of their nonperturbative aspects. In particular the duality [1],[2] between type II strings on Calabi–Yau threefolds and heterotic string on K3, leading to supersymmetric theories in four dimensions, makes possible the exact determination of string theory spacetime instanton effects, reducing to the exact field theory result of [3] after taking appropriate limits [4]. The underlying duality (at present) is however now understood as the duality between Ftheory in 8 dimensions on K3 and heterotic string on [5][6]; fourdimensional type II/heterotic duality than follows from further fibration over , using variants of the adiabatic argument introduced in [7].
Alternatively one can get theories with minimal supersymmetry in four dimensions by fibering the eight dimensional duality such as to obtain a Calabi–Yau fourfold on the Ftheory side and a CalabiYau threefold on the heterotic side [5][8]; Calabi–Yau fourfold compactifications of Ftheory have been discussed recently in [9],[10][11]. While the geometrical data of the compactification manifolds are largely fixed by the adiabatic arguments, the choice of the appropriate vector bundle on the heterotic side  the generalization of the choice of the instanton numbers in K3 compactifications  is not yet known in general, given X on the theory side. Independently of this question one might ask to what extent more refined geometrical data of the fourfold  such as period integrals and the correlation functions calculated by the topological field theory [12]  will descend to relevant physical quantities of the compactification. To address this question it is then natural to attempt to take advantage of the previously detailed studies of dual pairs by choosing fourfolds obtained as fibrations of threefolds over a further .
It is useful to think about the various dual descriptions as obtained from limits of two–dimensional compactifications. Specifically, after compactification on , Ftheory on is dual to Mtheory on and after further compactification on we have a duality between Ftheory on and type IIA on , which is the valid view for the discussion of periods and mirror symmetry in a geometrical string theory compactification on the fourfold. There are two particularly interesting limits to consider starting from this theory: first we can undo the or compactification by taking special limits in the Calabi–Yau moduli space. In this case we go back to the fourdimensional theories, e.g. heterotic string on . The second is to take the large base space limit of . In this case one flows to a theory which looks locally like in four dimensions, e.g. heterotic sting on K3 times the extra torus; in fact we will see that one obtains precisely the periods in this limit. It is suggestive to think about the world sheet instantons associated to the base departing from the large base space limit as breaking corrections. In sect. 2 we discuss the behavior of the fourfold periods in the large base limit.
The derivation of the period integrals and the correlation functions of the topological field theory rely on methods of mirror symmetry between Calabi–Yau fourfolds. A concept of mirror symmetry for Calabi–Yau folds for has been defined in [13] for one moduli cases^{†}^{†} For a discussion of mathematical aspects of mirror symmetry see [14][15].. For the more complicated fourfolds which are relevant in the present context, we develop the appropriate framework in terms of toric geometry in sect. 3, defining the fundamental correlation functions of the topological field theory directly in terms of the period integrals and a natural ring structure present in the toric variety. Other then in and in the one moduli cases considered in [13], the 3pt functions calculate a whole set of invariants , counting the Euler number of the moduli space of rational curves subjected to constraints on the location of the curves in the manifold, which arise from operators associated to codimension 2 submanifolds in .
In the second part we apply these methods to elliptically fibred fourfolds which are fibrations of Calabi–Yau threefolds which have itself wellknown heterotic duals in four dimensions. In sect. 4 we determine the correlation functions and the invariants associated to them and describe the geometrical meaning of the Kähler moduli which relates them to the moduli of the heterotic dual. In sect. 5 we make some verifications on the numbers calculated in the fourfold by imposing appropriate constraints and comparing the result with the known numbers of rational curves in the fibre. We discuss some interesting properties of the couplings and their role in the spacetime effective theory.
We then turn to the question of whether a superpotential is generated in the fourdimensional supersymmetric Ftheory compactification. In sect. 6 we analyze possible wrappings of fivebranes in an Mtheory compactification, following [8]. A complete classification of appropriate divisors (sixcycles) is given using intersection theory on the toric hypersurface; it turns out that a superpotential is indeed generated generically. We then ask about the physical effects related to these superpotential terms. As might be expected from the conjectured duality to heterotic string on a threefold, tensionless strings and compactifications of them play an important role. In sect. 7 we investigate singularities in the complex structure moduli space related to fibration singularities and possible gauge symmetry enhancement and describe the geometrical properties of the relevant divisors which provide the link to physical properties. In some cases the instanton generated superpotential can be interpreted as world sheet instantons of the magnetic noncritical string in six dimensions. Special singularities which appear in the moduli space when the five brane intersects or coincide with tensionless strings from three branes wrappings are discussed in sect. 8. A new kind of theories arises if nonperturbative effects generate a everywhere nonzero tension for the string with “classically” zero tension. In this case one obtains in the appropriate scaling limit a conventional field theory, however with a hidden string at a nonperturbatively generated low energy scale related to the scale of supersymmetry breaking.
2. Periods on the fourfold
One of the first questions about mirror symmetry of fourfolds and its use to determine nonperturbative effects in Ftheory compactifications is, which kind of nonperturbative effects are expected to be treated by the topological sigma model and which kind are not. In threefold compactifications mirror symmetry allows to determine the exact Kähler moduli space of the type IIA theory on from the map to the complex structure moduli space of the type IIB theory on the mirror . From the brane point of view this is possible since the complex structure moduli are associated to 3cycles on , however there is no twobrane available in type IIB which can be wrapped on these 3cycles to generate an instanton effect. Therefore the classical computation in the type IIB theory is exact and using the mirror map one obtains information about the world sheet instanton corrected Kähler moduli space of the type IIA theory on . The same can not be said about the other moduli space  of the type IIB theory on or of the type IIA theory on  since the latter theory has Dirichlet two branes which do generate complex structure moduli dependent instanton effects. Moreover the string coupling constant is a hypermultiplet and there are perturbative corrections in the type IIA string theory.
We will be primarily interested in the Kähler moduli space of type IIA compactified on the fourfold , including the corrections to the correlation functions calculated by the isomorphisms of the two topological theories, called the and the model. It would be interesting to know possible factorization properties of the full nonperturbative moduli space, a problem which is of course closely related to a similar question about (0,2) moduli spaces. Generally we expect that different than in the three dimensional case there are corrections that are not taken into account by conventional mirror symmetry based on the isomorphism of twodimensional topological theories. However the information provided by the exact mirror map should be enough to pin down the individual origin of an instanton effect (thus counting Dbranes states ) from the scaling behavior whereas the exact contribution will contain an additional sum of corrections as e.g. in the case of D2 brane instantons in type IIA theory [16]. Moreover it is an interesting question, what is the freedom that is not fixed by the holomorphic bundle structure starting from the apparently rather complete information on the type IIA side. We will see later that the answer is related to the the cohomology of the fourfold, which is special in many respects.
In the following paragraph we consider Ftheory compactification on fourfolds obtained from fibering elliptic Calabi–Yau threefolds over a twosphere, , of volume . If has a K3 fibration in addition to the elliptic fibration, this theory is expected to have a heterotic dual by fibrewise application of the 8 dimensional duality between Ftheory on K3 and heterotic string on [5].
2.1. Periods in the large base space limit
It is instructive to consider the large base space limit of Ftheory on ; in this case one expects to recover supersymmetric IIA on in four dimensions or the dual representation, heterotic string on . Since we want to use mirror symmetry to extract physical couplings from the integrals over the holomorphic form on a CalabiYau manifold it is useful to make precise this limit on the period integrals.
The observables of the model on the mirror manifold are in correspondence with elements of the middle cohomology of , , or rather a subspace of it in the case of fourfolds as will be discussed in more detail in the next section. Mirror symmetry relates the correlation function of the model on to those of the model on . In the model observables are associated to cohomology elements in or equivalently dimension homology cycles . The relation between the periods of the fourfold and the periods of the threefold fibre can be best understood in this last representation.
In the model on the threefold , the periods^{†}^{†} A prime refers to the fibre data in the following. are related to the 0 cycle , 2cycles , 4cycles and one 6cycle in . If we fibre over a to get a fourfold , we can think of the elements as obtained from joining elements with 0 and 2 cycles in the base . In this way we get homology cycles of the fourfold, generating the socalled vertical primary subspace of .
In the model on , expanded in special coordinates around a large complex structure point, the periods in special coordinates are of the form
where with the prepotential. Note that in the model the leading classical terms^{†}^{†} That is powers of the rather than instanton corrections of the periods can be interpreted as the volume of the homology cycles. In the large base space limit world sheet instanton corrections from the base are suppressed and integrating over the homology cycles of the fourfold reduces to an integration over the threefold cycles , possibly multiplied by the classical volume of the base , if is obtained from by joining the whole base. The periods of the fourfolds in the large base space limit are then simply given by combining these factors with the threefold result (2.1):
From the definition of the homology cycles on it is clear that nonvanishing intersections involve only pairs of elements which intersect on ; more precisely there is a set of cycles with and a set of cycles with with nonvanishing intersection only between and cycles and the intersection form given by that of the Calabi–Yau fibre. This implies in particular that the Kähler potential of the fourfold reduces that of the threefold compactification plus a constant
Starting from the precise relation in the large basis limit of the fourfold periods of the compactification and the structure of a compactification on the threefold fibre, it is suggestive to treat the instanton corrections associated with the base space modulus as breaking deformations of a theory.
Note also that the periods of the fourfold (2.2) are algebraically dependent; this is not only true in the large base space limit but simply a consequence of
which provides a nontrivial algebraic relation between the entries of the period vector. This is different than in the odddimensional case, where the first nontrivial equation derived from (2.3) involves a derivative acting on one and leads to a differential equation relating the periods.
If instead of fibering the threefold over a we consider a fourfold of the type , eq. (2.2) becomes exact. Let be the elliptically fibred threefold with base ; there is a point in the moduli space with the appearance of tensionless strings [6]. Then we have precisely the same situation as in [17], where a torus compactification of this string is considered, leading to in four dimensions. In this fourfold compactification the gauge coupling is determined from the Calabi–Yau periods in the usual way (taking into account the Ftheory limit); moreover it is easy to see from the results in [18] that the relevant periods at the tensionless string point are precisely those over the shrinking del Pezzo inside , implying its appearance in the final result of ref. [17].
3. Mirror map and Yukawa couplings
The description of moduli spaces of dimensional Calabi–Yau manifolds in terms of a holomorphic section of the Hodge bundle and period integrals over this holomorphic form has been given in [19],[20]. The concept of a mirror map relating npoint functions of and type topological field theories associated to a dimensional Calabi–Yau manifold and its mirror has been defined in [13], see also [21]. In this section we provide the general framework for the description of fourfolds with an arbitrary number of moduli in terms of toric geometry.
3.1. Toric description of and
Batyrev [22] has given a construction of mirror pairs of ddimensional Calabi–Yau manifolds as hypersurfaces in (d+1)dimensional toric varieties , where and denote the reflexive polyhedra defining the combinatorial data of and . We will use this description of Calabi–Yau fourfolds in the following.
Let denote the integral vertices of . The toric variety contains a canonical torus with coordinates . Then is defined as the zero set of the Laurent polynomial
where the coefficients are parameters characterizing the complex structure of . In [22][23]^{†}^{†} See also [24]. Batyrev shows that the Hodge numbers are determined by the polyhedron data as
where denotes faces of and the dual face of . and are the numbers of integral points on a face and in the interior of a face, respectively.
If the manifold has holonomy rather than a subgroup, then and the remaining nontrivial hodge number is determined by [9]. The Euler number is .
The target space toric variety can be described as
a generalization of projective space with scaling symmetries acting on the coordinates of and a disallowed set which consists of the unions of intersections of coordinate hyperplanes as determined by the so called primitive collections (see e.g. [25],[26]); e.g. for ordinary projective space one has and .
Apart from the combinatorial data , a specific phase of the Calabi–Yau manifold depends on the choice of a regular triangularization of . This defines in turn a choice of set of generators for the relations between the integral vertices^{†}^{†} Here we restrict ourselves to the set of vertices which lie on edges or faces of ; in the general case it can be necessary to consider also vertices corresponding to those interior points on faces of which represent automorphisms of [26]. , , called the the Mori vectors . The Kähler cone of the mirror is then the dual of the cone generated by the Mori generators. Starting from these data one obtains a system of differential equations, the PicardFuchs system, for the periods over the holomorphic form on [27]. The period integrals on are then given as linear combinations of the solutions to the PicardFuchs system.
There is a natural ring structure on from taking unions and intersections of toric divisors . The intersection ring is defined as the quotient ring , where is the ideal generated by linear relations and a set of nonlinear relations ; the latter is called the StanleyReisner ideal and determines the disallowed set .
In the next section we will relate the elements of at degree (where is here the complex codimension of a homology element) to observables of the type model on ; here is the so called primary vertical subspace of [13] which is the subspace of generated by wedge products of elements in . The ideal determines the dimension of the ring at degree ; in fact, for Calabi–Yau fibered fourfolds one has for .
Another distinguished set of generators of is determined by the divisors as defined by the Kähler cone of . Let be the forms dual to the special flat coordinates on the Kähler moduli space, centered at a large radius structure limit of maximal unipotent monodromy. Let be the Kähler form, and the divisors dual to the . We can use equivalently as generators of the intersection ring . In particular, if is the intersection form of
where the convention is that is the value of the integral , then the top element of dimension 4 of is simply while the volume of is obtained by replacing the divisors by the coordinates in (3.3) and relaxing the condition on the summation indices in (3.3).
Other topological invariants of are defined by integrating elements of wedged with the Chern classes of , :
with the obvious index structures. For holonomy, [9].
3.2. The model
Mirror symmetry implies that the correlation functions of two topological field theories defined on a Calabi–Yau manifold and its mirror are isomorphic. The correlation functions of the first theory, the model defined on , depend on the complex structure (CS) moduli of in a purely geometrical (classical) way and can be calculated straightforwardly. On the other hand the correlation functions of second theory, the model defined on , depend on the Kähler moduli (KM) of in a complicated way due to the presence of worldsheet instanton corrections. Mirror symmetry allows to determine these A model correlation functions by construction of the explicit mirror map from the complex structure moduli space of the B model to the Kähler moduli space of the A model.
Choice of a basis for the A model To match the moduli space of the CS moduli space of the model to the Kähler moduli space of the model we first chose a basis in the model in the following way. The basis for the primary vertical subspace with the most natural geometrical interpretation is given by forms Poincare dual to submanifolds of complex codimension [12]. Specifically we will chose a basis generated by the (1,1) forms dual to the special coordinates on the KM space, , and wedge products of them:
As mentioned above the dimension of the basis is reduced by the intersection properties of the dual homology elements determining the range of the lower index of the coefficients . E.g., in the case of Calabi–Yau fibrations, where , the intersection of the divisor dual to the base with itself is empty, , implying that there can appear at most one power of in the definition of the (this is the same kind of argument that ensures the linear coupling of the Kähler coordinate identified as the dilaton in K3 fibrations). In general the are chosen such that the elements generate the degree subspace of .
Topological metric, operator product expansions and correlation functions The 2pt functions define the flat metric on in terms of integrals of the basis elements over :
In fact the metric is constant in the flat variables [28] and nonzero only for pairs of operators with .
The r.h.s. is then determined by the coefficients of a given basis (3.5) together with the intersection numbers .
The factorization properties of the topological field theory ensure that all correlation functions can be expressed in terms of the fundamental 2pt and 3pt functions. Similarly as in the case of threefolds there is only one independent type of 3pt functions, namely , which contain the full information about the moduli dependence
The are determined in terms of the operator product coefficients , :
to be
While the 2pt and 3pt functions are the fundamental objects of the underlying topological theory, the simplest object on the fourfold which can be defined entirely in terms of the marginal operators are the 4pt functions
whose classical piece is given by the intersection numbers of . Factorization in terms of 2pt and 3pt functions yields
where we use a matrix notation . Nontrivial conditions on the ring coefficients follow from associativity of the operator products:
where the second identity follows from the first using . This identity provides highly nontrivial relations between the instanton corrected correlation functions.
3.3. Basis for the model and the mirror map
The next step to find the mirror map is the construction of a basis of the observables of the model which matches the properties of the above chosen basis for the model:
The appropriate basis for the model can be defined [13] using the Gauss–Manin connection , the flat metriccompatible connection on the Hodge bundle over the CS moduli space . The following construction is a generalization of the procedure in [13]; we can therefore focus on the complications introduced by the higher dimensional moduli space as compared to the one moduli case considered in [13]. In particular we will define the 3pt functions directly in terms of the period integrals over the holomorphic form and the intersection ring .
The fundamental step in the construction of the model basis in [13] is the replacement of the operator product involving a charge one operator with the action of the unprojected GaussManin connection
where now denote the basis elements of the model and the directional derivative is defined in terms of the parametrization of the deformations corresponding to marginal operators by the special flat coordinates . This definition implies a holomorphic dependence of the basis as opposed to the other natural choice, a basis of elements of pure type . We will now define a basis matching the property (3.10) of the model basis using the intersection ring and a map [26], where are the logarithmic derivatives with respect to the complex structure moduli of the model [27]
Let be a basis of topological homology cycles spanning the primary horizontal subspace and be the dual cohomology elements fulfilling
Furthermore let be a set of elements of , where are the holomorphic bundles of forms with antiholomorphic degree at most , fulfilling
The following relations are elementary:
Comparing (3.14) with (3.6), (3.7) it follows that is the basis matching the properties (3.10) provided that
We now construct the basis in two steps: a) First chose a basis for the . Of course we have . The are then obtained by choosing independent generators^{†}^{†} We use here to denote the element in that corresponds 11 to the operator . of the ring at degree and defining
where are differential operators of degree obtained from the map which follows from by transformation to the basis. The topological metric in this basis is then given by . b) We have to find the basis of cycles that satisfies (3.13) with the given basis . This can be done by fixing the leading logarithmic behavior of the period integrals
the exact periods are then determined by the solution of the PicardFuchs system with the appropriate leading behavior.
At the large complex structure point of the CS moduli space , the solutions to the Picard–Fuchs system have the leading behavior , where stands for any of the CS moduli . In fact there are precisely solutions with leading behavior , as a consequence of the relation between the intersection ring and the ring of differential operators [27][26]. The basis with the property is then fixed by the condition
where the ellipsis denote terms involving powers with and polynomial corrections. For convenience we state explicitly the expression for the leading piece of , as obtained from (3.16) by trivial matrix multiplication:
where is the intersection form given in (3.3), and is the Poincare dual of (as is obvious from the relation ). The exact expressions are then determined as the linear combination of the solution to the PicardFuchs system with the appropriate leading behavior. For more details we refer to app. C.
3pt functions All the fundamental 3pt correlators are then determined explicitly in terms of the period integrals on the middle dimensional cohomology of the Calabi–Yau 4fold . Namely, from
and (3.13) we obtain the final formula for the Yukawa coupling :
where with leading behavior .
Integrating the relations (3.17) over the cycles we obtain the PicardFuchs equation satisfied by the periods:
where hatted indices are not summed over. This is the holomorphic form of the differential equation reflecting the restricted Kähler structure of the CS moduli space of the fourfold. The corresponding linear system is the system obtained by integrating (3.17) over the manifold.
Finally note that the full intersection matrix of the period vector is obtained from the topological metrics as
3.4. Counting of rational curves
One of the most striking aspects of the calculation of the worldsheet instanton corrected 3pt couplings of the model on a Calabi–Yau threefold via mirror symmetry is the interpretation of the integral coefficients of the expansion in terms of the number of rational curves of multidegree on [29]:
The factor in the denominator of (3.20) takes into account the contribution of multiple coverings. This interpretation has been justified in the framework of the topological sigma model in [30]. The generalization of these argument to the case of dimensional Calabi–Yau manifolds has been given in [13]; in particular it was shown that the multiple covers contribute in the analogous way as in three dimensions.
On the other hand, the additional factor in (3.20) gets modified. Let us recall the relevant fact of the definition of the correlation functions in the topological field theory. By definition the local operators have delta function support on maps with the property ; here is the 2d world sheet and a codimension homology cycle of the Calabi–Yau target space . For the case of a operator, is a divisor, in fact in our choice of basis one of the divisors as defined by the Kähler cone of . The factors arise from the multiple intersections of a curve with that divisor and count the degree of the curve with respect to it In the case of the 3pt functions on the fourfolds there are two charge one operators involved, so one gets to factors of the relevant degrees, and , of the curve [31].
The third operator has charge two and is associated to a codimension two homology cycle  in our basis a linear combination of intersections of two Kähler divisors . Differently from the previous case the condition is a real constraint on the curve ; we have to adjust the position of the curve in the manifold to satisfy this condition. As a consequence the numbers appearing in the 3pt functions do count the appropriate Euler number of the moduli space of rational curves subject to a constraint. The constraint varies with the choice of and thus with the choice of . Therefore we do not expect to get the same numbers from 3pt functions involving different operators .
This is actually a nice circumstance for the present case of Calabi–Yau fibrations. In general, rational curves of the fiber get moduli in the fourfold from “moving them over the base”. Therefore the Gromov–Witten invariant of a curve in the threefold is generically not the same as the invariant of the same curve in the fourfold. However we will show that one can always fix the curves by choosing the appropriate operators ; in this case the numbers of the fourfold indeed coincide with those of the Calabi–Yau fiber. The appropriate choice for the is clear: one of the intersecting divisors will be the Calabi–Yau fiber itself  tautologically this imposes no constraint on the curves of the fiber. The second divisor which we intersect with to obtain a codimension 2 cycle plays the role of the third charge one operator in the threefold calculation and contributes naturally another factor of , counting the number of intersections of and in the fiber.
4. Threefold fibered fourfolds: toric construction and calculation of invariants
In the following we use the above construction to analyze some examples of fourfolds with four Kähler moduli which have at the same time phases which allow elliptic, K3 and Calabi–Yau threefold fibrations. This structure will allow various interpretations in terms of compactifications of theory, type IIA/IIB and heterotic/type I theories. In particular much is known about the theories on elliptic threefolds with base which can be fibered to fourfolds with four Kähler moduli, one from the base and three from the Calabi–Yau fiber . In addition to the type of threefold fiber there is the freedom to chose the bundle structure involving the base ; this will determine in particular the Calabi–Yau threefold of the dual heterotic theory. In table 1 we collect examples of fourfolds with four moduli arising in this way; the precise definition in terms of reflexive polyhdedra is given in the text and in appendix D.
Schematically the fibration structure is of the form indicated in the first column of table 1. The three complex dimensional base of the elliptic fibration can be thought of a collection of three factors with nontrivial bundle structure. The Chern classes of these bundles are described by the vector (a,b;c). Here the first two entries refer to the bundle structure of the “top” (the base of the elliptically fibered K3) over the other two factors, and the third one to the structure of the remaining over the base^{†}^{†} For a consistent notation, a choice has been made for those cases, where the base of the threefold fibration can be chosen in different ways..

In the following columns we give the hodge numbers and the Euler number. and denote the number of deformations which can not be realized as polynomial deformations in the toric description; they will play a role later on. In general there is more then one Calabi–Yau phase; in this case the stated fibration structure is present at least in one of those phases.