hep-th/0009228 IASSNS-HEP-00/73 UTTG-04/00 DUKE-CGTP-00/19

Codimension-Three Bundle Singularities in F-Theory

Philip Candelas, Duiliu-Emanuel Diaconescu, Bogdan Florea,

David R. Morrison, and Govindan Rajesh

Mathematical Institute, University of Oxford,

24-29 St. Giles’, Oxford OX1 3LB, England

School of Natural Sciences, Institute for Advanced Study,

Einstein Drive, Princeton, NJ 08540, USA

Theory Group, Department of Physics, University of Texas at Austin,

Austin, TX, 78712, USA

Center for Geometry and Theoretical Physics, Duke University,

Durham, NC 27708, USA

We study new nonperturbative phenomena in heterotic string vacua corresponding to pointlike bundle singularities in codimension three. These degenerations result in new four-dimensional infrared physics characterized by light solitonic states whose origin is explained in the dual F-theory model. We also show that such phenomena appear generically in Higgsing and describe in detail the corresponding bundle transition.

September 2000

1. Introduction and Overview

A remarkable achievement of string theory in recent years consists of understanding various nonperturbative effects associated to the breakdown of worldsheet conformal field theory. An example which has received much attention in the literature is the small instanton singularity in heterotic string theories [1]. These singularities occur in the context of heterotic string theories compactified on a K3 surface and are associated to the simplest pointlike degenerations of the background gauge bundles. Such degenerations have been shown to result in nonperturbative effects in six dimensions which can be understood either in terms of D-brane physics [1–5], or more generally, from the point of view of F-theory [6–9]. Various nonperturbative aspects of four dimensional heterotic strings have also been studied in detail [10–19]. The common feature of all these effects is that they can be ultimately related to the six dimensional small instanton singularity by an adiabatic argument. They have been accordingly interpreted in terms of heterotic fivebranes wrapping holomorphic curves in the Calabi–Yau threefold. From a mathematical point of view, such CFT singularities correspond to codimension-two bundle degenerations, precisely as in the six dimensional situation.

In this paper we consider a new class of nonperturbative effects specific to heterotic compactifications on Calabi–Yau threefolds. The singularities treated in the present work are qualitatively new, being associated to codimension-three bundle degenerations. This is a novel class of degenerations which have not been studied in physics so far and which are specific to four dimensional compactifications. As such, we expect qualitatively new infrared effects in four dimensional theories which will be discussed below.

The main tool for analyzing these singularities is heterotic/F-theory duality which encodes the bundle data in the geometry of a singular Calabi–Yau fourfold. This gives a pure geometric interpretation of the perturbative heterotic spectrum and determines at the same time the nonperturbative massless spectrum associated to CFT singularities. Generically, pointlike bundle singularities are expected to result in a certain type of spacetime defect where the nonperturbative degrees of freedom are localized. While this is also the case here, the nature of the resulting defect is very hard to understand. This is caused by our poor understanding of codimension-three degenerations of solutions to the Donaldson–Uhlenbeck–Yau equation. In particular, no explicit throat-like supergravity solution is known in this case.

In order to gain some insight into the nature of these singularities, it may be helpful to highlight the most important physical aspects by comparison with the small instanton transition. The bundle acquires pointlike degenerations which can be regarded as three dimensional defects filling space-time. However, there are no such stable excitations in the bulk M-theory, therefore such a defect is effectively stuck to the nine dimensional wall. This fact makes its physical properties quite obscure since it is not clear how to identify the light states governing the dynamics.

The F-theory picture is however more explicit. As expected, the bundle singularities correspond to special points on the F-theory base where the elliptic fibration develops certain non-generic singularities. These are superficially similar to the singularities occurring in the F-theory presentation of small instantons. So one might think by analogy that each such defect would correspond to a blowup of the three complex dimensional base. In fact, this is not the case since it will be shown in section two that the smooth fourfold obtained by blowing up the base is not Calabi–Yau. This is in good agreement with the absence of a ‘Coulomb branch’ noted previously (since the size of the exceptional divisor would be related to a displacement of the defect in the M theory bulk, which is forbidden).

Quite remarkably, it turns out that in the present case, there exist Calabi–Yau resolutions involving only fiber blowups. Recall that the resolution of the typical ADE singular fibers occurring in F-theory consists of a chain of two-spheres with specific intersection numbers in agreement with the corresponding Dynkin diagram. On top of each point in the base we have generically such a collection of spheres. A careful analysis reveals the fact that above the special singular points the resolved fiber contains an entire complex surface, i.e., a manifold of dimension four rather than a collection of two-spheres. Even more surprising is the fact that the occurrence of this surface is basically automatic; no extra blowups are necessary and there are no extra generators of the Kähler cone.

This has interesting consequences for physics, which are easier to understand by compactifying the four dimensional F-theory model down to three dimensions on a circle of radius . According to standard duality, this is equivalent to M theory on the resolved fourfold, the size of the elliptic fiber being proportional to . The presence of the surface in the fiber results in new light degrees of freedom in the low energy spectrum. We can have a string corresponding to the M fivebrane wrapped on the surface and a tower of particle states arising by wrapping membranes on holomorphic curves in . We regard the nonperturbative massless excitations as a sign of a singularity in the heterotic CFT. However, at the present stage it is very hard to get more insight into the low energy dynamics.

After this outline of the physics, let us describe next the precise context in which such singularities may be encountered. It is a common fact in string theory that singularities of various sorts are associated to phase transition between string vacua. As discussed in more detail later, it turns out that the pointlike singularities considered here appear generically in the context of the Higgs phenomenon in F-theory. More explicitly, we consider a typical transition corresponding to a family of singular Calabi–Yau fourfolds with generic fiber singularity which is enhanced to along a subspace of the moduli space. Technically, such an enhancement is realized by setting to zero certain parameters of the Weierstrass model. When this apparently simple transition is studied in detail one notices the presence of extra codimension-three singularities and the nonperturbative phenomena described above.

We can get a new perspective on this transition by making use of the spectral cover construction of Friedman, Morgan, and Witten [20]. At generic points in the moduli space we have a smooth holomorphic bundle of rank three. At the transition point, this bundle degenerates in a controlled manner to a singular object which is technically a coherent sheaf. Coherent sheaves have made their appearance in a number of places in physics. For example, in the linear sigma model approach to modes [21], the monad construction of the gauge bundle often results in a reflexive coherent sheaf [22] rather than a bundle. However, at least in the examples studied in [22], reflexive sheaves define non-singular CFT’s without the exotic phenomena described above. Given the fact that singularities in string theory tend to have a universal local behavior, it is reasonable to assume that this is the generic behavior.

In fact, this is consistent with the spectral cover description of the transition. We will show in section three that, at the transition point, the bundle degenerates to a non-reflexive rank three sheaf. Moreover, this sheaf admits a natural local decomposition as a sum of a rank two reflexive sheaf and the ideal sheaf of a point. After the transition, the heterotic vacuum will be described accordingly as having two distinct sectors. We have a perturbative CFT part corresponding to the reflexive rank two sheaf, which gives an gauge group and a certain number of matter multiplets. The second sector consists of nonperturbative degrees of freedom localized at certain points in the Calabi–Yau threefold, and corresponds to the ideal sheaves. This is quite similar to the small instanton effects in six dimensions, one of the main differences being the absence of a Coulomb branch.

This concludes our brief overview of pointlike bundle singularities in string theories. More details and explicit constructions are presented in the next sections. We discuss the transition in F-theory, and explain the occurrence of the surfaces in section two. The heterotic picture, based on the spectral cover approach, is presented in section three. Some technical details are postponed to an appendix.

2. F-theory

2.1. Generalities

Our starting point for the F-theory description is a Calabi–Yau fourfold which is dual to the heterotic string on a Calabi–Yau threefold with a certain gauge bundle. The unbroken gauge group then appears as a singularity in the elliptic fiber of the Calabi–Yau fourfold.

We will choose the heterotic Calabi–Yau threefold to be elliptically fibered over a base , which we choose to be the ruled surface ; this threefold has Hodge numbers . The dual Calabi–Yau fourfold is then elliptically fibered over a threefold base which can be viewed as the total space of the projective bundle , where , and is some effective divisor in the base , related to the class describing the heterotic bundle by the relation [23,13,14]. Similar models have been considered in a different context in [24].

The fourfold is described in the vicinity of a section by the Weierstrass equation

within the bundle . We have and , the normal bundle of in , and we denote by one of its sections, i.e., . The geometry of the split singularity over the section (which corresponds to gauge group) [25] is then encoded in the following expressions for , and the discriminant :

where are sections of certain line bundles over : , , , and .

The fourfold described by (2.1) can be resolved to a nonsingular Calabi–Yau fourfold, in which the singularities of split type are replaced by rational curves whose intersection matrix reproduces the Dynkin diagram of (generically). No surprises are encountered during this resolution. One way to describe the resulting F-theory model is as a limit from three dimensions: first, compactify M-theory on the nonsingular Calabi–Yau fourfold, then consider the limit in which all fiber components introduced during the resolution of the Weierstrass model acquire zero area (leading to enhanced gauge symmetry), and finally, take the F-theory limit by sending the area of the elliptic fibers to zero, opening up a new effective dimension.

The transition (un-Higgsing) to unbroken gauge group is described in F-theory terms by the condition , which results in the singularity being enhanced to fibers. Let us describe in some detail the geometry of this singular fourfold.

Under the condition , the discriminant actually has a factor of :

leading to fibers. The component of the discriminant is defined by the equation ; this latter equation is a cubic equation in whose discriminant with respect to is given by:

Let denote the locus , therefore the matter curve is . Then, it is easy to see that the locus is precisely . Finally, by and we denote the loci and respectively. The singularity type is enhanced to over the matter curve , which is the intersection locus of with the nodal part of the discriminant, . More interesting, the vanishing orders jump to over the intersection locus . (In the familiar case of an elliptic surface, this would be the signal that the Weierstrass model was not minimal. However, in the present context there is no birational change which can be made which would reduce those orders of vanishing.) The set where the vanishing orders jump to is precisely the singularity set of the corresponding heterotic sheaf (which would otherwise be a bundle, were it not for the presence of this locus). There is a cusp curve inside , which projects onto the curve in . The geometry of the singular fourfold is sketched in Figure 1.

Fig. 1: Geometry of the singular fourfold.

As explained in the introduction, resolution of this locus results in the appearance of entire complex surfaces over the locus in . This phenomenon is most efficiently observed by performing a weighted blowup of the Weierstrass model, which we now proceed to describe.

2.2. The Weighted Blowup

The weighted blowup is performed by introducing an additional variable , and assigning weights as follows:

We now rewrite (2.1) as a homogeneous degree 5 equation in the variables as follows

This is the weighted blowup of (2.1). In the patch , it is equivalent to (2.1), but when , we get

Note that over the point , this vanishes identically. Thus parametrise a family of hypersurfaces in except at , when we obtain all of . This is precisely the complex surface mentioned before. Its occurance is a direct consequence of the vanishing orders of the discriminant along .

The weighted blowup is really only the first stage of a complete toric
resolution of the singularity, and thus does not introduce any extra
Kähler classes beyond those already needed for the usual resolution of
singularities.
We present some further evidence for this lack of additional Kähler
classes by studying an explicit toric
example in the next subsection.^{†}^{†} It is also possible to perform
an explicit local resolution using the technique of
[26], which would be related by generalized flops to the resolution
presented in this section.

2.3. Toric Example

We now construct an explicit toric model of a Calabi–Yau fourfold elliptically fibered over base (i.e., , where and are the classes in ), with a section of split singularities. This fourfold is dual to heterotic strings compactified on the Calabi–Yau threefold which is elliptically fibered over , with an bundle with , and .

From index theorems and anomaly cancellation (see, for instance [13,14]), we expect the following Hodge numbers, .

The Calabi–Yau fourfold may be constructed as a hypersurface in a toric variety following the prescription of [13,14]. The dual polyhedron , which encodes the divisors of the polyhedron has vertices:

Standard toric methods [13,14] give, for the Hodge numbers of the fourfold, , indicating that the model has a background -flux turned on. Moreover, it is possible to find a triangulation of the polyhedron consistent with its elliptic fibration structure, such that each of the top dimensional cones has unit volume, guaranteeing smoothness of the corresponding Calabi–Yau fourfold. We assert that this polyhedron gives the F-theory dual of the heterotic vacuum described above. , in agreement with expectations. Note that the Euler characteristic is not divisible by

We can now study the effect of un-Higgsing the unbroken gauge group to . The heterotic bundle is now , with , and unchanged. Index theorems and anomaly cancellation predict, for the dual Calabi–Yau fourfold, Hodge numbers .

The dual polyhedron describing the Calabi–Yau fourfold has vertices

The fourfold has the following Hodge numbers, in accordance with our expectations: . Once again, it is possible to find a triangulation of the of polyhedron consistent with its elliptic fibration structure, such that each of the top dimensional cones has unit volume, guaranteeing smoothness of the corresponding Calabi–Yau fourfold.

It should be emphasized here that no extra Kähler classes other than the ones corresponding to the resolution of the locus are present in the fourfold. Since we expect, on general grounds, that the resolution of the singularity yields an entire complex surface over specific points in , we conclude that the appearance of the complex surface in the resolution of the locus does not introduce any extra Kähler classes.

The corresponding F-theory model has some features whose physical effects are difficult to explain in detail. We begin as before in three dimensions, with M-theory compactified on the nonsingular Calabi–Yau fourfold. When we allow the rational curves in the fibers to shrink to zero area, again we get enhanced gauge symmetry, but this time there are surfaces shrinking to points as well as curves shrinking to points. Wrapping the M-theory fivebrane on such surfaces suggests that the spectrum should contain light strings, while wrapping the M-theory membrane on curves within such surfaces would produce a tower of light particle states. All of these states are presumably present as well in the F-theory limit.

2.4. Comparison with Codimension-Two

It is worthwhile making a comparison between the geometry of these codimension-three singularities, and the analogous phenomenon in codimension-two. In the latter case, the F-theory interpretation of a small instanton singularity is that the total space of the elliptic fibration has acquired a singularity which can be resolved by a combination of blowing up the base of the fibration and blowing up the total space [6]. (It is the blowup of the base which leads to an additional branch of the moduli space.) In the codimension-three case, however, blowing up the base is not possible, because it destroys the Calabi–Yau condition.

To see this, consider first a simple model of the codimension-two phenomenon, represented by the Weierstrass equation

One of the coordinate charts when blowing up the base is , ; in that chart, the Weierstrass equation becomes

To make this new Weierstrass equation minimal, we must also change coordinates in and , using , . Our final Weierstrass equation is then

(This two-step change of variables corresponds to the two-step geometric process of a blowup and a flop which was used in [6] to describe this transition.)

According to the Poincaré residue construction, the holomorphic three-form was originally represented by

where represents the partial derivative of (2.7) with respect to the variable (which is not present in the numerator). In the new coordinate system (the minimal model of the blowup) this becomes

Since this latter is the Poincaré residue representation of a holomorphic three-form for the blown up threefold (2.9), our original three-form has acquired neither a zero nor a pole during this process. Thus, both threefolds can be Calabi–Yau and there is a transition between them. (Note that even though we only made the computation in a single coordinate chart, the order of zero or pole of the holomorphic three-form would be the same in any coordinate chart, so this is actually a complete argument.)

By contrast, let us make a similar computation for a fourfold, starting from the Weierstrass equation

We can represent one of the coordinate charts of the blowup by , , ; in that chart, the Weierstrass equation becomes

We again get a non-minimal Weierstrass model, which can be made minimal by the further coordinate change , . Our final Weierstrass equation is then

The holomorphic four-form was originally represented by

and in the minimal model of the blowup this becomes

Since this is times the Poincaré residue representation of a holomorphic four-form for the blown up fourfold (2.14), our original four-form has acquired a zero along the exceptional divisor . Thus, at most one of these two fourfolds can be Calabi–Yau (i.e., have a non-vanishing holomorphic four-form), and there is no physical transition between them.

3. Singular Bundles and Transitions

3.1. Spectral Data

We begin with a short review of the spectral cover approach to bundles on elliptic fibrations [20]. Let be a smooth elliptic Calabi–Yau variety with a section . As usual, we assume that is moreover a cubic hypersurface in , where is the anticanonical line bundle of the base.

According to [20], the moduli space of rank semistable bundles with trivial determinant on a smooth elliptic curve is isomorphic to the linear system , where is the origin of . This construction works for families of elliptic curves as well, if the singular fibers are either nodal or cuspidal curves. For the Weierstrass model introduced before, this yields a relative coarse moduli space which is isomorphic to the relative projective space . If is a rank bundle whose restriction to every fiber is semistable and regular with trivial determinant, then determines a section . Such a section is uniquely given by a line bundle over , and sections , .

The converse is not true, i.e., a section does not uniquely determine a bundle . Friedman, Morgan, and Witten [20] construct certain basic bundles associated to a section together with an integer . The construction is rather involved and it will not be reviewed here in detail. After some work, it can be shown that it is equivalent to the standard spectral cover construction [20]. Namely, the section determines a spectral cover which belongs to the linear system , where . In order to construct , let us consider the following diagram [20]