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Lie Groups, Calabi-Yau Threefolds, and F-Theory

The F-theory vacuum constructed from an elliptic Calabi-Yau threefold with section yields an effective six-dimensional theory. The Lie algebra of the gauge sector of this theory and its representation on the space of massless hypermultiplets are shown to be determined by the intersection theory of the homology of the Calabi-Yau threefold. (Similar statements hold for M-theory and the type IIA string compactified on the threefold, where there is also a dependence on the expectation values of the Ramond-Ramond fields.) We describe general rules for computing the hypermultiplet spectrum of any F-theory vacuum, including vacua with non-simply-laced gauge groups. The case of monodromy acting on a curve of A_even singularities is shown to be particularly interesting and leads to some unexpected rules for how 2-branes are allowed to wrap certain 2-cycles. We also review the peculiar numerical predictions for the geometry of elliptic Calabi-Yau threefolds with section which arise from anomaly cancellation in six dimensions.


Introduction
The F-theory vacuum [1] constructed from an elliptically fibered Calabi-Yau threefold X with section determines an effective theory with N = (1, 0) supersymmetry in six dimensions. Such supersymmetric theories will have fields in hypermultiplets, vector supermultiplets and tensor supermultiplets. (See, for example, [2] for a discussion of such theories.) For any particular F-theory vacuum, the taxonomy of the supermultiplets may be derived from the geometry of X as an elliptic fibration via seemingly straightforward methods in the case of the vector and tensor multiplets [3,4]. The classification of the hypermultiplet content has always been a little harder to carry out. Many methods have been proposed which allow the hypermultiplets to be determined from the geometry in certain cases [5][6][7][8][9][10][11][12][13].
The purpose of this paper is to outline a systematic approach to the problem of determining the gauge symmetry and hypermultiplet content of a given six-dimensional theory obtained from F-theory. (Note that as far as the moduli space of hypermultiplets in concerned, our methods utilize the associated type IIA compactification and thus also apply directly to the compactification of M-theory on X giving an N = 1 theory in five dimensions and to the compactification of the type IIA string on X to yield an N = 2 theory in four dimensions, provided that the expectation values of certain Ramond-Ramond fields have been tuned appropriately.) The methods we employ will not be particularly new but we will see that the process of analyzing the gauge group and matter content can be quite a bit more subtle than had previously been appreciated. In particular, the case of monodromy of the fibration leading to non-simply-laced Lie algebras requires some care. A particularly awkward case which has caused some confusion is when a Z 2 monodromy acts on a curve of A even singularities, i.e., a curve of I odd fibers in F-theory language. In this paper we resolve this problem in agreement with an observation by Intriligator and Rajesh in [14] concerning anomaly cancellation.
In section 2 we will show how many features of a Lie algebra structure arise naturally from an elliptically fibered Calabi-Yau threefold. This will allow us to elucidate the method for determining the gauge algebra. In section 3 we discuss exactly how to analyze the hypermultiplet content in the cases where the associated curves and surfaces within the Calabi-Yau threefold are smooth. We discuss the cases where these curves and surfaces are singular in section 4. This section includes some unexpected rules we are forced to adopt for 2-brane wrapping. Although the results of this section are less rigorous than the preceding section, we are able to give precise results in many instances which can be extended to the general case under the fairly conservative assumption that the relevant physics is determined locally from the geometry of the singularities.
Finally in section 5 we emphasize the peculiar numerical predictions which arise from anomaly cancellation in the F-theory compactification on X.

Lie algebras and Calabi-Yau threefolds
We begin with a Calabi-Yau threefold X which admits an elliptic fibration π : X → Σ, where Σ is a complex surface, and also assume that this elliptic fibration has a section. 1 The type IIA string compactified on X yields an effective four-dimensional theory with N = 2 supersymmetry; its strong-coupling limit, known as "M-theory compactified on X," yields an effective five-dimensional theory. One more effective spatial dimension is obtained in a limit in which the areas of all components of the elliptic fibers shrink to zero-this is the "F-theory limit." See, for example, [3,18] for an explanation of this.
We point out that most of the following analysis does not really depend upon this elliptic fibration structure and applies to M-theory and type IIA compactifications of X. We use the F-theory language as an organizational tool to give examples later on. One also has the advantage in F-theory of being able to use anomaly cancellation as a powerful tool in checking the consistency of results concerning spectra of massless particles. In the F-theory context we can freely exchange the notion of, say, an I n fiber and an A n−1 singularity. The former is the elliptic fibration description for the latter. Recall [4] that this is because although I n is really the extended Dynkin diagram of A n−1 and that one always ignores the components of the fiber which hit the chosen section of the elliptic fibration. 2 Thus, in the zero-area fiber limit of F-theory, a shrunken I n fiber gives the same physics as an A n−1 singularity one dimension lower.
Whenever rational curves in X are shrunk down to zero size we expect 2-branes of the type IIA string wrapped around these curves to contribute massless particles to the spectrum. It is precisely these massless states which are the focus of our interest in this paper.
Actually we need to be careful with the statement that massless states appear automatically when a brane wraps a vanishing cycle. There is always the subtlety of B-fields and R-R fields which should be tuned to the right value (usually denoted "zero" by convention) to really obtain a massless state. As emphasized in [19] the relevant parameters to worry about in this context are the R-R fields. We may see this as follows. If one considers the type IIA string compactified on X then deformations of the Kähler form (and B-field) on X are given by vector moduli. Suppose we use these Kähler moduli to shrink down a holomorphic 2-cycle to obtain an enhanced gauge symmetry. Once we reach this point of enhanced symmetry we may have a phase transition releasing new hypermultiplet degrees of freedom. Thus at the point of phase transition, these new parameters, which include R-R fields, are fixed at some value. Reversing this point of view, we may tune parameters in the hypermultiplet moduli space to achieve an enhanced gauge symmetry but these parameters include R-R fields. Thus we need to assume always that the R-R parameters have been tuned to the appropriate values required to obtain the enhanced gauge symmetries we discuss below.
Witten [6] analyzed how to determine the massless particle content for a given configuration of rational curves. Let us assume that a given rational curve lives in a family parametrized by a moduli space M. In the simplest case one has an embedding M ×P 1 ⊂ X. An isolated rational curve is a trivial example of this where M is simply a point.
According to Witten's calculation, one half-hypermultiplet may be associated to the fact that a 2-brane breaks half of the supersymmetry. This half-hypermultiplet is then tensored with the total cohomology of M in an appropriate sense. The result is that if M is a point, then we simply obtain a single half-hypermultiplet. If M is an algebraic curve of genus g then we obtain a single vector multiplet and g hypermultiplets. This was also argued by a different method in [5]. Note that for any wrapping we may also wrap with the opposite orientation to double this spectrum.
Of central interest to us is the fact that compactifying a type IIA string theory (and thus M-theory and F-theory) on a Calabi-Yau threefold X produces a theory with a Yang-Mills sector. The gauge fields may be viewed as arising from integrating the R-R 3-form of the type IIA string over 2-cycles in X to produce 1-forms. 3 These 1-forms play the role of the Yang-Mills connection. In addition, the 4-cycles in X which are dual (via intersection theory) to these 2-cycles will play an important role. Let F denote the 4-form field strength of the R-R 3-form in the type IIA string. Note that the 2-branes of the type IIA theory are electrically charged under this field-that is for a 6-dimensional shape M 6 (such as a six-sphere) enclosing the seven directions transverse to a fundamental 2-brane. Upon compactification we will be wrapping 2-branes around a 2-cycle in X to produce a point particle in four-dimensional space-time. To find the charge of this resulting particle we may take M 6 = S 2 × S i , where S 2 is a sphere in four dimensional space-time enclosing the particle and S i is a 4-cycle within X.
It follows that in the type IIA compactification 1. We have b 2 (X) = b 4 (X) gauge symmetries of the type U(1), each labelled by an element of H 4 (X), in addition to the U(1) gauge symmetry coming from the R-R 1-form (whose charge is measured using M 6 = X, the generator of H 6 (X)).
2. If a 2-brane wraps a 2-cycle C a to produce a particle then the "electric" charge of this particle under the U(1) symmetry associated to a 4-cycle S i will be the intersection number (S i ∩ C a ).
We thus obtain a perturbative U(1) b 4 (X)+1 gauge symmetry in type IIA. In the M-theory compactification, there is no R-R 1-form, and the eight transverse directions to the M-theory 2-brane are enclosed by M 7 = S 3 × S i , so the total "perturbative" gauge symmetry is given by U(1) b 4 (X) . In the F-theory limit, the only 4-cycles which contribute gauge fields are those with intersection number zero with the elliptic fiber; moreover, 4-cycles which are the inverse images of 2-cycles in the base Σ are associated to tensor multiplets rather than gauge fields. Thus, in the F-theory limit, we get a "perturbative" gauge symmetry group of As is now well-known, and as we will discuss, the wrapped 2-branes will elevate this U(1) b 4 (X)+ε gauge symmetry to a non-abelian Lie group (since certain wrapped branes include vector multiplets in their spectra), where ε = 1, 0, or −b 2 (Σ) − 1 for IIA, M-theory or Ftheory, respectively. From now on we will concern ourselves only with the Lie algebra of the gauge symmetry. It was noted in [20] that, at least in F-theory, the global structure of the gauge group may be recovered from the Mordell-Weil group of X as an elliptic fibration. 4 If this u(1) ⊕(b 4 (X)+ε) appears as the Cartan subalgebra of our gauge algebra then the discussion above implies that we may make the following identifications. Let h be the (real) Cartan subalgebra, let h * be the dual space, and let Λ ⊂ h be the coroot lattice and Λ * ⊂ h * be the weight lattice so that the Cartan subgroup U(1) b 4 (X)+ε is naturally identified with h/Λ. For the IIA compactification, we take Λ = H 4 (X, Z) ⊕ H 6 (X, Z) and Λ * = H 0 (X, Z) ⊕ H 2 (X, Z), and in M-theory, we take Λ = H 4 (X, Z) and Λ * = H 2 (X, Z). In F-theory, we begin with the orthogonal complement within H 4 (X) of the elliptic fiber E, and then we mod out by π −1 H 2 (Σ) (that is, Λ = [E] ⊥ /π −1 H 2 (Σ) ⊂ H 4 (X)/π −1 H 2 (Σ)); we then take Λ * = Hom(Λ, Z) to be the dual lattice of Λ. In each case, a 2-brane wrapped around a particular 2-cycle is then naturally associated with an element of the weight lattice and its charges under the Cartan subalgebra are given in the standard way.
We work this out in detail in several particular cases. Consider first the case that X contains a "ruled" complex surface S admitting a fibration π : S → M, for some M, where all fibers are isomorphic to P 1 . The fibers will shrink down to zero size in the F-theory limit. The simplest example of this is M × C 1 where M is a Riemann surface of genus g and that C 1 ∼ = P 1 is in the fiber direction. That is to say, in our elliptic fibration π : X → Σ we have a curve M ⊂ Σ over which the fiber is I 2 . Clearly we have massless states appearing for the 2-branes wrapped around C 1 . We also have a u(1) symmetry associated to S 1 ∼ = M × C 1 . Let us consider the normal bundle of a single C 1 curve. This normal bundle may be written as O(a) ⊕ O(b) where a + b = −2 by the adjunction formula and the fact that X is a Calabi-Yau space. Since this curve may be translated along the M direction one of these line bundles must be trivial. Thus the normal bundle is O ⊕ O(−2) where the O(−2) describes the normal bundle direction which is also normal to S 1 . Therefore (S 1 ∩ C 1 ) = −2. This tells us that we have a vector supermultiplet and g hypermultiplets from wrapping 2-branes around M × P 1 all with charge −2 with respect to the u(1) gauge symmetry associated to this divisor. Similarly by wrapping with the opposite orientation we obtain a copy of this except with charge +2.
These vector supermultiplets enhance the u(1) symmetry to su(2) in the usual way and we have an additional g hypermultiplets in the adjoint representation. The key point is to notice in this construction that the condition has played the role of the Cartan matrix of su(2). 5 The next simplest case is where we have a set of curves C 1 , . . . , C n which may intersect each other and are each isomorphic to P 1 and lying in the fiber direction. We assume that M ×( i C i ) embeds algebraically into X. 6 We now have a Cartan matrix given purely by the configuration of C 1 , . . . , C n . Applying the above method we obtain the standard F-theory result of a simply-laced enhanced gauge symmetry as listed, for example, in table 4 of [18].
As noted first in [11] the real power of this Cartan matrix approach is that it gives a clear way of describing non-simply-laced gauge algebras. Consider a less trivial example of 5 Of course there is a sign difference here compared to usual Lie algebra theory. This sign difference is purely due to the convention that Lie algebra theorists insist on the Cartan matrix being positive definite, rather than negative definite. If string theory had been studied before Lie algebras then the sign would be the other way! 6 We can consider more generally a situation where we glue together n distinct P 1 fibrations over M along appropriate disjoint sections, forming a chain. In the remainder of this paper, we will continue to explain by example and will not explicitly state the most general form of the algebraic surfaces which contract to M in the F-theory limit. ruled surfaces as shown in Figure 1. In this example the moduli space M 1 of the curve C 1 is different from the moduli space M 2 the curve C 2 . Think of M 1 as the vertical direction in the figure. We obtain ruled surfaces S 1 = M 1 × C 1 and S 2 = M 2 × C 2 . We have a two-fold cover M 2 → M 1 branched at one point in Figure 1. Any other branch points are not shown in the figure.
The intersection matrix of this configuration may be written This is the Cartan matrix for sp(2) (or so (5)) and so our enhanced gauge algebra should be sp (2). This phenomenon of obtaining a non-simply-laced symmetry algebra was first noted in [7] inspired by the construction of [21]. There it was explained by monodromy acting on the fibers as follows. Let M 1 be embedded in Σ and let M 2 be a two-fold cover of M 1 (branched at various points). Over a generic point in M 1 we see that, ignoring the component meeting the chosen section, the Kodaira fiber consists of one line from S 1 and two lines from S 2 forming the (dual) Dynkin diagram of su(4). Moving along a closed path in the complement of the set of branch points of M 2 → M 1 we will exchange the two lines in S 2 . This action on the Dynkin diagram is induced by an outer automorphism of su(4) and the invariant subgroup under this outer automorphism can be taken to be sp(2).
One might therefore suspect that the effective gauge algebra is the monodromy-invariant subalgebra of the simply-laced gauge symmetry generated locally by the vanishing cycles. This was the assertion in [7]. Unfortunately it is an ambiguous statement. 7 Let us analyze carefully all possible outer automorphisms of SU(2k). An element g of SU(2k) satisfies ( Tḡ )g = 1. Complex conjugation t : g →ḡ is an example of an outer automorphism. Indeed this acts on the Dynkin diagram of SU(2k) by reflection about the middle node. Clearly the invariant subgroup under this outer automorphism is given by g real. But this yields the group SO(2k)-not what we were expecting! A general outer automorphism of SU(2k) can be obtained by combining complex conjugation with an arbitrary inner automorphism, yielding g → h −1ḡ h, where h ∈ SU(2k) (there are no other possibilities since that would imply further symmetries of the Dynkin diagram). Since this outer automorphism acts on the Dynkin diagram as the reflection, it is also a viable candidate for the monodromy action on the gauge group. In this general situation, the invariant subalgebra satisfies The case h = 1 yields SO(2k) as stated before. Now if we put where I is the k × k identity matrix, we obtain the group Sp(k)-as desired. In this case, the outer automorphism is an involution, but this is not a requirement in general. We see then that the method of directly working out the Cartan matrix from intersection theory is a better way to determine the effective gauge algebra in F-theory than trying to find subalgebras invariant under outer automorphisms. The latter method is ambiguous. One might try to assert that F-theory picks out the "maximal" invariant subalgebra under all possible outer automorphisms. Indeed, sp(k) is "bigger" than so(2k) in as much as it has a larger dimension (although in general so(2k) ⊂ sp(k)). However, even this approach is inadequate as will be shown by examples in section 4.
Note that in the M-theory or type IIA compactifications, an ambiguity of the sort we have discovered is actually to be expected. As we have already pointed out, if the Ramond-Ramond fields have non-zero expectation values then some of the non-abelian gauge fields will become massive; when these are integrated out, the gauge group becomes smaller. This is precisely what happens when the outer automorphism of the covering Lie group is varied in the construction above. The gauge algebra which we wish to determine is the one in which these effects have been turned off so that the F-theory limit can be taken. (A similar phenomenon of variable gauge group depending on the precise value of an outer automorphism has been observed in a closely related context by Witten [23], and applied in [24,25]).

Counting hypermultiplets
In the last section we described how to determine the gauge algebra in F-theory (or Mtheory or IIA string theory) by determining the Cartan matrix from intersection theory. Similar methods will in principle determine the hypermultiplet spectrum completely as we now discuss.
First there can be the case of a family of rational curves acquiring extra rational curves at certain points in the family. In the context of elliptic fibrations this can be seen as collisions of curves in Σ over which there are singular fibers. The simplest example is a transverse collision of I n and I m .
The resolution of singularities associated to this collision was explained in [26], and applied to the case of an I n -I 1 collision in the context of string theory in section 8.2 of [11].
The key point is that there exist rational curves within the collision with normal bundle O(−1) ⊕ O(−1). One of these curves C is the intersection of two ruled 4-cycles, one lying over the curve of I n fibers, and the other lying over the curve of I m fibers. The normal bundle of C in X is naturally the direct sum of the normal bundles of C in each of these 4-cycles, and each of these is O(−1). Thus this curve appears as (minus) a fundamental weight. The above rules imply that we have found a curve representing the (lowest) weight of the (n, m) representation of su(n) ⊕su(m). By adding other (possibly reducible) curves in the collision fiber we may indeed build up the full (n, m) representation. Thus the transverse collision of a curve of I n and I m fibers yields a hypermultiplet in the (n, m) representation of su(n) ⊕ su(m). (This same result had earlier been determined using quite different methods in [27]. Another approach which is closer to ours appeared in [10].) Many other "collisions" can be explained in similar ways. However, if the extra rational curves at the collision point have normal bundles other than O(−1) ⊕ O(−1), then Witten's calculation does not directly apply. General methods for evaluating the corresponding contribution to the hypermultiplet spectrum are not known.
The case of non-simply-laced symmetry algebras raises even more complicated possibilities. Some of the hypermultiplet matter can appear in a somewhat "non-local" manner as we now explain. Suppose we are in a situation analogous to Figure 1. Let us consider the example of a type I 2k fiber (where we again ignore the component passing through the chosen section). Let the middle component in the chain have a moduli space given by M 1 and the other components have moduli space M 2 where M 2 → M 1 is a double cover. That is, we have a Z 2 -monodromy acting on the I 2k fiber (in the only possible way). Figure 1 is the case k = 2.
According to [6] we should obtain g(M 1 ) hypermultiplets for 2-branes wrapping the middle component and g(M 2 ) hypermultiplets for 2-branes wrapping each of the other components. Note that g(M 2 ) ≥ g(M 1 ) from the double cover. There are additional hypermultiplets arising from wrapping connected unions of these components. In fact, each of the positive roots of the covering algebra su(2k) is represented by such a connected union, some of which are fixed by the monodromy, and others of which are exchanged in pairs under the monodromy. The ones which are fixed under monodromy have M 1 as moduli space, while those which are exchanged in pairs have M 2 as their moduli space.
When we organize these weights in terms of representations of sp(k), we find that the invariant subspace describes the adjoint of sp(k) while the anti-invariant subspace describes the remaining weights in the adjoint of su(2k). On the other hand, each invariant positive root contributes to the invariant subspace, while the roots exchanged in pairs contribute to both the invariant and anti-invariant subspaces. We conclude that the adjoint of sp(k) occurs g(M 1 ) times while the weights in the anti-invariant subspace each occur g(M 2 )−g(M 1 ) times.
We demonstrate which weights appear in the example of k = 2 in Figure 2. We show the weights of the adjoint representation. The dots represent weights associated to M 1 , i.e., from C 1 . The circles represent weights associated to M 2 , i.e., from C 2 . It is important to note that reducible curves may also be wrapped by 2-branes. That is, two rational curves intersecting transversely at a point may be viewed together as a nodal rational curve. These wrappings  of reducible curves are required to obtain all the adjoint weights of the vector multiplets of the previous section. The reducible curve C 1 + C 2 has moduli space given by M 2 -since C 2 has a moduli space given by M 2 . Looking at Figure 1 we see that there is also a chain of rational curves in the class C 1 + 2C 2 but note this this combination is invariant under the Z 2 monodromy and so has moduli space given by M 1 . These circles form the weights of 5 of sp (2). Ignoring the zero weights for now we see that the adjoint appears g(M 1 ) times and the 5 of sp(2) appears g(M 2 ) − g(M 1 ) times. Indeed the zero weights also work out correctly. A zero weight must represent an uncharged hypermultiplet and therefore a modulus. We may use the work of Wilson [28] to demonstrate this. Wilson showed that a Calabi-Yau threefold containing a ruled surface M × P 1 has a moduli space which preserves this ruled surface only in codimension g(M). That is, there are g(M) deformations of the Calabi-Yau threefold which destroy this ruled surface. Applying this to both ruled surfaces, we get g(M 1 ) + g(M 2 ) deformations. On the other hand, each sp(2) adjoint contains a two-dimensional weight zero eigenspace while each 5 contains a one-dimensional weight zero eigenspace. Thus the dimension of the weight zero eigenspace is 2(g(M 1 )) + (g(M 2 ) − g(M 1 )), which simplifies to g(M 1 ) + g(M 2 ), as claimed.
The above construction may be easily generalized to sp(k): 8 Theorem 1 Let Z 2 monodromy act on an I 2k fiber in F-theory so that the central component of the fiber has moduli space M 1 and the outer components have moduli space M 2 . Thus M 2 → M 1 is a double cover. Then the resulting gauge algebra is sp(k) and we have g(M 1 ) hypermultiplets in the adjoint representation and g(M 2 ) − g(M 1 ) hypermultiplets in the Λ 2 representation (which has dimension k(2k − 1) − 1).
Similarly e 6 with Z 2 monodromy will yield an f 4 gauge algebra with g(M 2 ) − g(M 1 ) hypermultiplets in the 26 representation (in addition to the usual g(M 1 ) adjoints). Also so(2k) with Z 2 monodromy will yield an so(2k − 1) gauge algebra with g(M 2 ) − g(M 1 ) hypermultiplets in the vector 2k − 1 representation. In the case of so (8)  One might note that the above cases with Z 2 monodromy may be combined into a simple rule as follows. Let s be the simply-laced Lie algebra which contains the actual gauge symmetry algebra g as a subalgebra invariant under an outer automorphism given by the monodromy action. (In fact, in each of the above cases, the outer automorphism which we use actually has order 2 as an automorphism of s, not merely order 2 as an automorphism of the Dynkin diagram.) We may then decompose the adjoint representation of s as follows where V − is a (possibly reducible) representation of g on which the generator of the Z 2 outer automorphism acts as −1. The above rules may be combined to say that we obtain g(M 2 ) − g(M 1 ) hypermultiplets in the V − representation. As we have already noted above in the case g = g 2 , the rule will be different if the monodromy group is not Z 2 . In fact, we will see more generally in section 4.1 that if the outer automorphism representing the monodromy has higher order, the simple rule expressed in equation (6) must be modified. In addition to these "non-local" hypermultiplets coming from rational curves moving in families one may also obtain further hypermultiplets from collisions of curves of reducible fibers as in the I n -I m collision discussed above. 9 Note that some simple collisions may just induce monodromy without further contributions (that is, their contributions are completely accounted for by the representation V − obtained in eq. (6) ). As an example we show in Figure 3 the generic case of a Spin(9) gauge symmetry in F-theory.
This figure shows an I * 1 fiber along a section of the Hirzebruch surface F n . This section has self-intersection +n and is denoted C 0 in the notation of [18]. In the most generic situation, the rest of the discriminant locus of the elliptic fibration will consist of I 1 fibers along curves which intersect C 0 as shown in cubic collisions 10 produce extra rational curves in the fiber but no monodromy while the 2(n + 6) transverse collisions produce monodromy but no extra rational curves. Thus the 2(n + 6) collisions produce n + 5 of the vector 9's of so (9). An analysis of the rational curves in the cubic collision shows that we have n + 4 spinor 16's. Assuming n = −4, the existence of these spinors shows that the gauge group must be Spin (9). This agrees perfectly with section 4.6 of [8]. Similarly all the other results of [8] may be confirmed. Finally in this section let us return to the case of Z 2 monodromy acting on a curve of su(2k) singularities to give an effective sp(k) gauge symmetry. We will consider the Higgs branch in which we give expectation values to the hypermultiplets so as to break completely this sp(k) gauge symmetry. Recall that the geometry of moduli spaces of supersymmetric field theories in question imply that the dimension of this Higgs branch should equal the total dimension of the representations of charged hypermultiplets minus the dimension of the gauge group which is broken. We will observe that the geometry is in accord with Theorem 1. We do this by describing the deformations after shrinking all of the curves in the fibers to zero volume. In section 4.2 we will use the ideas introduced here towards the justification of our Main Assertion stated in the next section, which states that the gauge algebra in the case of su(2k + 1) with Z 2 monodromy is sp(k).
We let π : M 2 → M 1 be an unramified (for simplicity) double cover of M 1 . In addition, we denote by ι : M 2 → M 2 the involution which exchanges sheets of the double cover. We now describe a local Calabi-Yau threefold X containing the geometry of su(2k) with Z 2 monodromy over M 1 . First we construct a Calabi-Yau threefold Y with an su(2k) fibration over M 2 without monodromy. Then X will be constructed as a Z 2 quotient of Y .
We construct a singular threefold inside the bundle V = K k M 2 ⊕ K k M 2 ⊕ K M 2 as the variety defined by the equation where x and y are in K k M 2 and z is in K M 2 . This threefold has an A 2k−1 singularity along M 2 , which is identified with the zero section of V . It has trivial canonical bundle by the adjunction formula. In a moment we will construct a nowhere vanishing holomorphic 3-form on it, giving independent verification of this fact. The desired threefold Y is obtained by blowing up the singular locus k times in the usual way to obtain a chain of 2k − 1 ruled surfaces over M 2 .
To obtain the desired geometry, we take the quotient of Y by the fixed point free involution obtained by using ι on M 2 while sending (x, y, z) to (y, x, −z). Note that the fibers of K M 2 over p ∈ M 2 and ι(p) ∈ M 2 are canonically identified with the fiber of K M 1 over π(p), so this map makes sense. Using the explicit description of the blowup and the fact that x and y are interchanged, it follows that there is Z 2 monodromy.
To show that the quotient X by this involution has trivial canonical bundle, it suffices to show that the involution preserves the holomorphic 3-form on Y . It suffices for our purposes to compute on the singular model. Let ω be any holomorphic 1-form on M 2 . Then is a holomorphic 4-form on V with values in K 2k+1

M
. Now, thinking of ω as a section of K M 2 , we divide by ω to obtain the nowhere vanishing 4-form on V with values in K 2k M 2 which is independent of ω. Finally, the residue is the holomorphic 3-form on the singular model of Y . It is clearly invariant under the involution. The deformations of X may be described as the deformations of Y in V which preserve the involution. The general deformation of Y (up to change of coordinates) is given by where the f i are sections of K i M 2 . Note that we are implicitly assuming g(M 2 ) > 0 to construct these deformations. The invariance condition is that f i lies in the (−1) i -eigenspace of ι.
4 The case of monodromy on su(odd).
While we appear to have given fairly general rules in the previous section for computing the massless particle spectrum of an F-theory compactification, there are actually many cases where the rules we have given so far become difficult to apply.
In particular, Witten's analysis of the moduli space of rational curves in [6] assumes that everything is smooth (and reduced). This need not be the case. We will discuss some awkward cases which appear quite commonly in F-theory.
We begin with a discussion of a case which has caused some confusion in the literaturethat of Z 2 monodromy acting on a gauge algebra of su(2k + 1). The approach of asking for the largest subalgebra invariant under an outer automorphism is not that helpful in this case. Putting h = 1 in eq. (4) shows that so(2k + 1) is one possibility. Decomposing 2k + 1 as k + 1 + k and putting (where I is the k × k identity matrix) shows that sp(k) is another possibility. (This form of h is nicely adapted to the action on the Dynkin diagram.) Now it so happens that dim(so(2k + 1)) = dim(sp(k)) (and that this is the largest dimension which can occur). So which is the gauge algebra that F-theory actually wants? Using the approach of section 2 we immediately run into a problem. One of the ruled surfaces, which we will denote S 1 , swept out by the reducible components of the fibers will look inevitably locally like the surface in C 3 . We show a sketch of (the real version of) this surface in Figure 4. Each line C 1 in this surface crosses another line C ′ 1 in the same class. In the case of su(2k + 1) for k > 1 there will be other smooth surfaces. This case is a little hard to visualize. In Figure 5 we show the case of monodromy acting on su (5). fact that S 1 is singular and from the figure read off the intersection matrix This would imply that the gauge algebra is su(3). In general, according to this argument, Z 2 monodromy acting on su(2k + 1) would produce su(k + 1)-neither of the possibilities suggested above! It would be the most obvious algebra suggested by "folding the Dynkin diagram up" by the outer automorphism. We could get an sp(k) Cartan matrix from the case in question if we could somehow tie C 1 and C ′ 1 in Figure 4 together. That is, somehow the rules of 2-brane wrapping would have to assert that C 1 may not be wrapped alone-one must also wrap the intersecting curve C ′ 1 simultaneously. Since S 1 is singular, there is no known reason for ruling such a possibility out. By considering the reducible curve C 1 + C ′ 1 as a single curve, we effective replace S 1 by a simple ruled surface. Thus we would reduce Figure 5 to Figure 1. That is, the case su(2k + 1) is reduced to su(2k) and so we get sp(k) under monodromy.
At this point therefore we do not really seem to know what the gauge algebra is. The geometry seems to suggest su(k + 1) or sp(k) while the outer automorphism argument suggests sp(k) or so(2k + 1). We will now give various arguments in support of the following assertion.
Main Assertion For an F-theory compactification on an elliptic threefold with a curve of I 2k+1 fibers (which locally suggests a symmetry of su(2k + 1)) with Z 2 monodromy, the resulting gauge symmetry is sp(k) (provided that the R-R fields are set to "zero"). This assertion corrects some statements which appeared in earlier literature where it had been assumed the resulting gauge symmetry was so(2k + 1) for reasons we discussed above. As mentioned in the introduction one can show that the spectrum of various F-theory models, such as point-like instantons on a D n singularity, is anomaly free for sp(k) but inevitably would have anomalies in some cases had the gauge algebra contained so(2k + 1). 11 Note that the outer automorphism of SU(2k + 1) which yields sp(k) actually has order 4, since its square is conjugation by the matrix where I is the k × k identity matrix; h 2 is not a central element of SU(2k + 1). This outer automorphism of course still induces the required reflection of the Dynkin diagram, as we explained near the end of section 2. This modifies the analysis which led to eq. (6) as follows. We let the outer automorphism of order 4 act on s and decompose into eigenspaces: each of which will be a representation of g (possibly reducible). As before, the eigenspaces for eigenvalues ±1 can be accounted for by certain positive roots which are left invariant under the involution and by certain pairs of positive roots which are exchanged under that involution. The moduli space for the former is M 1 and for the latter is M 2 ; when we consider the quantization of the D2-branes wrapped on the corresponding curves, we find 2g(M 1 ) halfhypermultiplets for each of the invariant roots, and 2g(M 2 ) half-hypermultiplets for each of the pairs. Since each pair contributes to both the +1 and −1 eigenspace, this adds up to a total of 2g(M 1 ) half-hypermultiplets in the adjoint representation of g, and 2g(M 2 ) − 2g(M 1 ) half-hypermultiplets in the representation V − .
We have yet to account for the representations V i and V −i . In fact, these are the roots which contain either C or C ′ , which-as we have argued above-cannot occur as wrapped D2-branes at the generic point of the parameter curve M 1 if we are to reproduce the Cartan matrix compatible with our Main Assertion (or at least, such wrapped branes cannot produce vector multiplets). 12 However, as we will see in section 4.1, the representations V i and V −i do occur in the hypermultiplet spectrum-perhaps because at the branch points of the map M 2 → M 1 , C and C ′ are identified and there is no apparent obstruction to wrapping the D2-brane there.
To be more concrete concerning the case at hand, with s = su(2k + 1), g = sp(k), and the outer automorphism determined by the h in eq. (13), we have the second exterior power of the fundamental of sp(k) (which has a trivial one-dimensional summand), and the fundamental representation of sp(k). Thus, the predicted spectrum is: • g(M 1 ) hypermultiplets in the adjoint representation • g(M 2 ) − g(M 1 ) hypermultiplets in the second exterior power representation (including its trivial summand), and • additional hypermultiplets in the fundamental representation.
In fact, an anomaly calculation [30] predicts that there will be precisely 2g(M 1 ) − 2 + 3 2 b = 2(g(M 2 ) − g(M 1 )) + 1 2 b such hypermultiplets. One possible interpretation of this formula is that there are two fundamentals (V i and V −i ) associated to the parameter curve M 2 and an additional half-fundamental at each branch point.

The case of su(3) → sp(1).
A Kodaira type IV fiber would intrinsically produce an su(3) gauge symmetry but monodromy may act on this fiber producing the case of interest. At first sight this might not look like such a good candidate for examination since su(2) ∼ = sp(1) ∼ = so(3)! However, the hypermultiplet spectrum will allow us to distinguish the cases. Consider the case of amassing point-like E 8 -instantons on an orbifold point of a K3 surface along the lines analyzed in [32]. We will be interested in the case of four instantons and six instantons on a C 3 /Z 3 quotient singularity. From result 3 and figure 7 of [32] we may deduce the spectrum without encountering any difficulties. In the case of four instantons, the Z 3 singularity may actually be partially resolved to a Z 2 singularity without affecting the particle spectrum. This Z 2 singularity may then be effected by a "vertical" line of I 2 fibers (in the notation of [32]). The six instanton case is effected by a vertical line of I 3 fibers. The results are • For four point-like E 8 -instantons on a Z 3 singularity we have a nonperturbative enhanced gauge algebra of su(2) with hypermultiplets in four 2 representations.
• For six point-like E 8 -instantons on a Z 3 singularity we have a nonperturbative enhanced gauge algebra of su(2) ⊕ su(3) ⊕ su(2) with hypermultiplets as (2, 1, 1) We may also produce exactly the same physics by using a vertical line of type IV fibers. The configurations of curves of Kodaira fibers in the base of the elliptic fibration is shown in Figure 6 for the cases of four and six instantons respectively. These diagrams are again similar to those presented in [32] and represent the situation after the base has been blown up the requisite number of times. The short curved lines represent fragments of the the curve of I 1 fibers.
Let us begin with the case of six instantons on the right of Figure 6. The lines of type II fibers produce no gauge symmetry enhancement. The upper and lower diagonal lines of type IV fibers each collide once transversely with a line of type II fibers and once nontransversely with the curve of I 1 fibers. Actually these collisions are very similar. 14 Each of these collisions produces Z 2 monodromy in the type IV fiber producing the geometry of Figure 4. Thus each of these diagonal lines of type IV fibers produce an su(2) (or sp (1)) gauge symmetry.
The remaining vertical line of type IV fibers collides with the other two lines of type IV fibers. Resolving this collision shows that no monodromy is induced. Thus this vertical line represents an su (3)  We are left with having to account for a hypermultiplet 2 in each of the su(2)'s. This must come from the monodromy-inducing collisions of the diagonal lines of type IV fibers with the lines of type II and type I 1 fibers. As these collisions are all the isomorphic locally, each collision must produce a half-hypermultiplet 2. This is in agreement with our comments concerning the V i and V −i representations at the end of the previous subsection. The collision point is the point around which the monodromy acts and so it associated with the location of the curve denoted D in Figure 4.
The choice of associating this su(2) as the k = 1 case of sp(k) or su(k + 1) differs as explained above by whether we view the positive root of su(2) as being associated to C 1 or to C 1 + C ′ 1 . Clearly in the latter case we have 2D = C + C 1 as divisor classes and so D naturally generates the 2 as required. If only C 1 were identified as the positive root then D would produce nothing new. Therefore we can only correctly identify the spectrum F-theory in the case of the geometry on the right-hand side of Figure 6 if we take one of roots of the gauge algebra to be C 1 + C ′ 1 . That is, there really does appear to be a rule in string theory which allows 2-branes to wrap C 1 + C ′ 1 together but not C 1 or C ′ 1 individually. We can further verify our picture by considering the spectrum for four instantons on the left of Figure 6. There are four collisions with the vertical line of type IV fibers, each producing monodromy. Thus, g(M 2 ) = 1 and there are b = 4 branch points. Following the arguments at the end of the previous subsection, we thus predict a spectrum consisting of one hypermultiplet in the Λ 2 C 2 ∼ = C representation (from V − ), and 2(g(M 2 ) − g(M 1 )) + 1 2 b = 4 hypermultiplets in the fundamental representation (i.e. V ±i ). This precisely agrees with the spectrum found above: there are four 2's of su (2). Even the V − representation "1" occurs correctly: it is the deformationà la Wilson [28] of M 1 , or in physical terms of the heterotic string, it is the deformation of the Z 3 singularity to a Z 2 singularity which does not affect the spectrum as noted earlier.
The rules of 2-brane wrapping are therefore rather unusual for the curves C 1 and C ′ 1 . As observed previously, away from the branch points, a 2-brane can never wrap C 1 or C ′ 1 individually. However, as we have just seen, at the branch points where C 1 and C ′ 1 coincide, the 2-brane is allowed to wrap the curve. In fact, when this wrapping is taken with both orientations, a hypermultiplet in the 2 of su(2) is produced for each branch point.
Since C 1 lies in the singular surface S 1 it is perhaps not surprising that the usual rules of 2-brane wrapping appear to break down. Anyway, since this same S 1 appears as the "end" component for the higher rank gauge groups of this type, assuming string theory wraps branes around curves in S 1 in a similar way in that context, we arrive at our Main Assertion.

Deformation to the su(even) case.
We can also give a different argument in favor of the sp(k) gauge group in case M 1 has positive genus. Let us start with the case of I 2k+2 fibers with Z 2 monodromy. By Theorem 1, this leads to a sp(k + 1) gauge group and at least one adjoint hypermultiplet. We will show in a moment that the corresponding A 2k+1 singularity can be smoothed to an A 2k singularity with Z 2 monodromy. This corresponds to giving a nonzero vev to a semisimple element of the adjoint hypermultiplet, and the sp(k + 1) gauge group gets Higgsed to some rank k subgroup g ⊂ sp(k + 1). We still have to determine what g without knowledge of the u(1) sp(k + 1) that acquires a vev. Clearly sp(k) ⊂ sp(k + 1) is possible, so we could have g = sp(k). We now argue that g = so(2k + 1) is impossible.
Proof This argument is due to R. Zierau. Suppose that there were an embedding of so(2k+1) in sp(k+1). Then the fundamental 2k+2 dimensional representation V 2k+2 of sp(k+1) would restrict to a representation of so(2k + 1), which necessarily decomposes as a fundamental representation V 2k+1 of so(2k + 1) plus a trivial representation. The alternating form on V 2k+2 restricts to a alternating form on V 2k+1 . Since V 2k+1 is odd dimensional, this form is degenerate. It's nullspace W ⊂ V 2k+1 is invariant under so(2k + 1), and is a proper subspace since the alternating form on V 2k+1 is not identically zero. This is a contradiction.
The singular surface in question given by eq. (14) may be written as and thought of as a double cover of the xz-plane branched along z = 0 and doubly along x = 0. It is the double branching that makes the surface singular. We may smooth the surface by deforming to Now the double branching has been split to x = 0 and x = ǫ. For a fixed value of z this process replaces a nodal rational curve by a smooth rational curve, where the nodal rational curve can be viewed as two rational curves intersecting transversely at a point. That is, each pair of intersecting lines in Figure 4 is replaced by a single line and the surface is smoothed. This smoothing process is remarkably benign at the level of global geometry. It is often possible to perform it even when the geometry of the ambient threefold, X, is completely smooth at all times.
We can then derive the sp(k) gauge symmetry indirectly as follows. The existence of the deformation shows that 2-branes are not allowed to wrap the individual lines of Figure 4. The deformation converts each pair of lines into a single new line. Thus if physics is not discontinuously affected by the deformation, the 2-branes contributing to vector particles must only be allowed to wrap the pairs of line in Figure 4 together. As we have observed in the discussion immediately preceding the Main Assertion, we can now conclude that we do indeed obtain sp(k). 15 There is of course a problem with the proof of our Main Assertion by this argumentthere may be global obstructions to such a deformation. Wilson's criterion suggests that such deformations only occur when g(M 1 ) > 0, where M 1 is the base of the fibration of Figure 4 as a ruled surface. We address this in part by giving an example of a deformation when g(M 1 ) > 0.
We return to the setup introduced at the end of section 3. Using the notation leading to eq. (11), we write the equation This gives su(2k + 1) with Z 2 monodromy at the generic point of M 1 . The deformation is simply To make sense of this, we have to say a little more about the blowup. The A 2k blowups are determined by a procedure given in [33] after choosing an ordering of the 2k + 1 factors of z 2k+2 + f 1 z 2k+1 + ǫf 2 z 2k = z 2k (z 2 + f 1 z + ǫf 2 ). Choosing the (z 2 + f 1 z + ǫf 2 ) factor to be in the middle, we obtain the desired geometry. The last blowup creates a single ruled surface, which smooths out the singular component. It is immediate to see from the description in [33] that for generic f i this is a smooth deformation of the desired type.
In this model, we have placed a restriction on the genus and have introduced localized matter at the zeros of f 1 . However, if we are willing to accept that the process of gauge symmetry enhancement is dictated by local geometry then this example is enough to justify our Main Assertion.

Numerical Oddities
Finally we close with a note on the peculiar numerical predictions dictated by anomaly cancellation in the six-dimensional physics produced by F-theory compactified on X. This has been discussed in many places before (for example [34]) and is often used as a method of enumerating the spectrum of hypermultiplets. Here we have outlined a systematic way of constructing the hypermultiplet spectrum and so the anomaly constraint becomes a peculiar numerical property of the geometry of an elliptic Calabi-Yau threefold.
For completeness we will repeat the anomaly condition here. We consider an elliptic fibration π : X → Σ with a section. Let G be the gauge group (or algebra) in six dimensions and ρ(Σ) the Picard number of Σ. Then anomaly cancellation along the lines of [35] yields the following where the hypermultiplets fall into representations R i of G and ε i is equal to 1 if the representation is real or 1 2 if the representation is complex or quaternionic (pseudoreal). Note that the trivial representations also contribute to the sum. These can be determined from the fact that the number of neutral hypermultiplet moduli are equal to h 2,1 (X) + 1.
As an example consider the extreme case of G ∼ = E 17 8 ×F 16 4 ×G 32 2 ×SU(2) 32 corresponding to 24 point-like E 8 -instantons on a binary icosahedral quotient singularity in the heterotic string [32]. The Calabi-Yau threefold for the F-theory description of this has ρ(Σ) = 194. Applying the methods of sections 3 and 4.1 to this threefold we also arrive at a spectrum of hypermultiplets of a (1, 2) ⊕ (7, 2) for each of the 32 copies of G 2 × SU(2). 16 These representations are quaternionic. Equation (24) The anomaly condition in eq. (24) has been verified in situations illustrating Theorem 1 and our Main Assertion (see the footnote in Section 4 of [14]).
Note that one may obtain further conditions from the anomaly cancellation condition. For example one may require the vanishing of coefficient of each "tr(F 4 )" term in the anomaly. See for example [9,34].
It would be very satisfying to give a purely geometric proof of eq. (24) and the other anomaly conditions. (A geometric proof which covers a wide variety of cases has recently been constructed [30].) Sadly at present the origin of this formula without using string theory is something of a mystery. Note that the existence of a section in the fibration π : X → Σ is necessary for this to work. If this requirement is not satisfied then there is no six-dimensional physics and the condition need not be satisfied [18] (for an example, see [36]).