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Branes, Bundles and Attractors: Bogomolov and Beyond

We discuss conjectures following from the attractor mechanism in type II string theory about the possible Chern classes of stable holomorphic vector bundles on Calabi-Yau threefolds. In particular, we give sufficient conditions for Chern classes to correspond to stable bundles.


Introduction
Let (M, g) be a compact complex n-dimensional Kaehler manifold, with Kaehler class J ≡

[g] ∈ H
(1,1) ∂ (M). Let V be a holomorphic vector bundle on M. The topological classification of such V is given by a K theory class, which up to torsion is determined by the Chern character ch(V ). We recall that ch(V ) is a sum of differential forms in H 2 * (M) ≡ ⊕ k H 2k (M, R), which given a connection with curvature F ∈ H 2 (M, End V ) can be written as ch(V ) = Tr e F Thus, its zero-form component is the rank of V , its two-form component is the first Chern class c 1 , and higher forms are polynomial in the Chern classes. We further recall that the bundle V is µ-stable if, for all holomorphic subbundles L, we have µ(L) < µ(V ), (1.1) where the slope µ(E) of a bundle E is defined by Note that a bundle is polystable if it is a direct sum of stable bundles with the same slope.
The same definitions can be made for a coherent sheaf V . In addition, given a submanifold These sets are known for curves and for a few surfaces. Even for the simplest surfaces, say M = P 2 , their precise structure is rather intricate [1]. Thus an approximate description, say by upper and lower bounds, is also of interest.
In this work we discuss the problem of characterizing these sets, and give some conjectures inspired by superstring theory. Let us state our main conjectures without further ado: In particular, there exists such a stable sheaf for any given ample class. In particular, there exists such a stable bundle for any given ample class.
The reason that stable reflexive sheaves arise here, along with a more detailed discussion of stable objects in string theory, will be given in section 3.2.
We will give a "physics proof" of these conjectures in section 4. Such conjectures arise in the study of compactification of superstring theory, in more than one way. The original problem of this type was that of constructing realistic vacua from the heterotic string [3]; see [4,5,6] and references there for the most recent work. Here, one seeks a holomorphic bundle V carrying a connection which solves the Hermitian Yang-Mills equations, g ab F ab = cI, (1.11) and satisfying the following topological constraints: the rank is 3, 4 or 5; the first Chern class is vanishing, and the second Chern class is determined by anomaly cancellation to be equal to that of the tangent bundle of M. By the DUY (Donaldson-Uhlenbeck-Yau) theorem, the question of whether such a bundle exists, is equivalent to asking whether CH contains an element satisfying these constraints.
Note that the topological conditions determine all but the third Chern class c 3 , which is unconstrained a priori. Thus, knowledge of CH would give us a list of allowed c 3 values.
Physically, such a value determines the number of families of quarks and leptons in four dimensions. Hence, a bound on the third Chern class in terms of the other data translates into a interesting bound on the number of families one can obtain in such compactifications.
A second physical problem, which will be of more primary interest in this paper, is that of characterizing the BPS particles in type IIa superstring theory, which arise from wrapping Dirichlet six-branes on M [7]. Loosely speaking, a Dirichlet six-brane is a seven real dimensional minimal volume submanifold of M × R 3,1 , where R 3,1 is Minkowski spacetime, carrying a vector bundle V with a connection satisfying the Yang-Mills equations. A BPS particle is the special case in which the embedding is a direct product of M with a timelike geodesic in R 3,1 , and the connection satisfies the Hermitian-Yang-Mills equations.
The Chern character ch(V ) determines the electric and magnetic charges of this particle, and thus we find a second physical interpretation of the set CH -it is the set of possible charges of BPS particles.
The definition of µ-stability, while precise, is not so easy to check, as we must know all holomorphic subbundles of V . Other necessary conditions for stability are known. Perhaps the most famous is the Bogomolov bound, which states that the Chern classes of a µ-stable holomorphic vector bundle V will satisfy This follows from the existence of a solution to hermitian Yang-Mills, and the positivity of the volume form.
For surfaces, the Bogomolov bound already gives a rough picture of CH (J). Consider the example of the projective plane P 2 ; the actual boundary of CH (J) as given in [8], is given by an inequality of the form Eq. (1.12) with a finite correction depending on µ.
Not much seems to be known for higher dimensions. It is known that any subset of CH (J) with fixed rank, c 1 and c 2 is finite [9,10].
How can the connections we discussed to string theory help us with this problem? Our approach will be based on the attractor mechanism [12,13,14]. This uses the relation between BPS particles and extremal black hole solutions of supergravity theory. The definition of a BPS particle in type II supergravity provides a map Z from the Chern character and Kaehler class to C, called the "central charge," [16] explicitly Note that the Kaehler form J has been promoted to B + iJ, the "complexified Kähler form" in H 2 (M, C).
Now, for a fixed ch(V ), one can consider Z as a function of B + iJ, and look for a local minimum of the quantity (1.14) By definition, the minimum ||Z min || is attained at an "attractor point" for V in complexified Kaehler moduli space. We refer to a ch(V ) for which ||Z min || > 0 as satisfying the "attractor condition," and let ATT ⊂ H 2 * (M) be the set of points satisfying this condition.
In [14], Moore conjectured that if a charge vector is in ATT , then there will exist a stable BPS particle with that charge. Specializing this conjecture to the case at hand leads to our basic claim: that ATT ⊂ CH . Determining ATT requires no knowledge about bundles and subbundles, only the de Rham cohomology (with its product structure) and Kähler cone of

M.
Working this out leads to a set of conditions on Chern characters which are sufficient for existence. The first of these will turn out to be Bogomolov bound Eq. (1.12). The next is a condition on the third Chern class: if V is stable and has vanishing first Chern class, and for some ample Kaehler classH, we find with an order one constant whose precise definition depends on stringy corrections; neglecting these it is 2 5/2 /3.
Let us now describe the attractor arguments in an intuitive way, referring to [14,17] for more details. Given a Chern character which satisfies the attractor condition, we want to see why a BPS particle with these charges must exist. This is because one can find a extremal black hole solution of supergravity with these charges, which to a distant observer is indistinguishable from a BPS particle. A more precise version of this argument relies on variation of parameters (the dilaton field) to interpolate between the black hole and the Dirichlet brane whose existence we are trying to determine. One can show that the existence of a BPS particle is independent of this parameter (since the dilaton sits in a hypermultiplet) The upshot is that the attractor condition is a sufficient but not necessary condition for ch(V ) ∈ CH (J), for all J in the basin of attraction of the attractor point, and thus for To get a candidate for a necessary condition, and better understand variation of stability, one must continue and discuss the far-reaching generalizations of the original attractor conjecture made by Denef in [17,18,19,20,21]. The original attractor argument assumed that the supergravity solution was spherically symmetric. This need not be the case; more complicated multi-center solutions provide examples of BPS bound states with charges which do not satisfy the the original attractor condition. In general, the existence of these solutions is determined by the existence of "split attractor" flows, this condition is manifestly dependent on J and has been shown to describe variation of stability in examples.
To the extent that variation of stability is always described by this mechanism, we would conclude that ATT ⊂ CH . It also suggests a prescription for how to enlarge ATT to better approximate CH , as we discuss in section 5.
In order to trust these physical arguments, we need two conditions to be satisfied. One of these, to justify the use of the large volume expression for the central charge Eq. (1.13) and the comparison with µ-stability, is that the attractor point satisfy J >> 1 in string units (more precisely, that world-sheet instanton corrections be negligible). We will discuss this condition later on; in principle the discussion could be generalized to include world-sheet instantons and the more general notions of stability introduced in [37,39].
The other condition, required so that supergravity is a good approximation to the more exact string theory description, is that the quantity ||Z min || 2 >> 1, as this is the condition that the area of the event horizon (and thus the curvature radius) be large compared to the Planck scale. A sufficient condition for this is for all non-zero components of the Chern character ch(V ) to be large.
More precisely, since Eq. (1.14) is homogeneous of degree two in ch(V ), for any given ch(V ), there will exist an N min such that for all N > N min the set of Chern characters obtained by the rescaling ch(V ) → Nch(V ) will satisfy this condition. Note that the definition of attractor point and the attractor condition are unchanged by such an overall rescaling.
We will refer to subsets of H 2 * (M, Z) or bounds on ch(V ) which are invariant under rescaling as "homogeneous." To the extent that the set of charges of BPS objects and therefore the sets CH can be understood from the genus zero attractor considerations we discuss, we conclude that these sets are homogeneous.
we see that it is homogeneous. Thus we raise the possibility that, along with our sufficient conditions, homogeneous necessary conditions on the Chern characters also exist. We discuss this in section 6.
Before making this physical discussion a bit more precise and work out the attractor bounds, we will in the next section, state our results and the related mathematical conjectures on the existence of stable bundles on Calabi-Yau threefolds and surfaces with ample canonical bundle and test them against hard mathematical theorems.
Note that the previous version of the paper contained a possible conjecture on a slight strengthening of Bogomolov inequality. However, as pointed out by M. Jardim, there exists examples which violate the stronger inequality. We will present an example due to M. Jardim in Appendix B.

Existing mathematical results
We begin this section by setting out some elementary features of the problem, and then briefly summarize some known mathematical existence theorems and constructions.
Let H ∼ = H 2 * (M, R) be the direct sum of the even de Rham cohomology groups. We consider it as a commutative algebra, with the standard cohomology product, and with the fundamental class M : H → R. Should we need it, a basis for H 2 (M, R) will be denoted . We can take these to be ample classes if we like.
Of course, only classes in H p,p (M, C) can be realized as the Chern characters of holomorphic bundles. Our primary interest is in M a Calabi-Yau threefold with h 1,0 = h 2,0 = 0, for which these span H.
The Mori cone MC ∈ H 2 (M, Z) is generated by effective curves in M. Its dual KC ∈ H 2 (M, R) of ample classes ω satisfying < ω, C > ≥ 0 ∀C ∈ MC is the Kähler cone.
Ample classes have various positivity properties [44]. We will use the following: consider H i with i = 1, 2, 3 ample on an irreducible complete threefold (such as M); then Given a sheaf V on M, we define the generalized Mukai vector γ(V ) ∈ H as This is of course simply related to ch(V ) by a linear change of basis, but turns out to emerge more naturally from the physics. One of its mathematical advantages is a simple statement of the Grothendieck-Riemann-Roch formula. Defining an involution on H which corresponds to dualizing a vector bundle, we have As this depends only on the Mukai vectors γ(V ) and γ(W ), we also write this as χ(γ(V ), γ(W )).
We now consider the subsets CH , CH and CH of the introduction, defined using the map (resp. CH and CH ).
Indeed, since the tensor product of two µ-polystable bundles (at a fixed J) is µ-polystable, each CH (J) is closed under multiplication -it is a subalgebra of H.

Homogeneous invariants
One way to simplify the problem is to discuss asymptotic results, valid for large Chern character. Thus, we define PCH , PCH and PCH to be the closure of the projectivized versions of the above; in other words v ∈ PCH if for any δ we have λv + ǫ ∈ CH for some λ ∈ R + and some ǫ ∈ H with |ǫ| < δ, etc.
One can define polynomials in the Chern classes which are invariant under an overall rescaling of ch(V ), and are thus well defined on PCH . A useful set of generators for these, introduced by Drezet, is defined by the following expansion, The ∆ i for i ≥ 2 are unchanged by tensoring with line bundles, and this can be used to infer the terms proportional to c 1 .
We furthermore have and in this sense the ∆ i linearize the algebra structure on H. Thus we have In particular, note that ∆ factorizes through For example, the Bogomolov bound is which is consistent with this. For n = 3, the set ∆CH (J) is determined by one more inequality, for some convex function f . For example, we might have f (x) = x a for any a ≥ 1.

Existence results
We consider the moduli spaces M J (r, c 1 , c 2 ) andM J (r, c 1 , c 2 ) of µ-stable vector bundles and sheaves respectively (with respect to J), with rank r and first and second Chern classes is not empty, we say that bundles of (resp. sheaves) of this topological type exist.
First, for a general smooth algebraic variety M of arbitrary dimension, an asymptotic result of Maruyama [9] states that bundles exist for r dim M 2 and c 2 (V )ω n−2 ≫ 0.
More precise statements are available for algebraic surfaces. An early general result where δ is a periodic function 1/2 δ 1. Actually, this result is sharp, that is, the imposed condition is also necessary if one excludes exceptional bundles, that is bundles with On a K3 surface, we have the following recent result by Yoshioka [22]: semistable sheaves (for a general ample class) exist if This is simply the requirement that the Mukai vector v(W ) = ch(W ) td (K3) satisfies Under the assumption that v is primitive, Yoshioka can actually show the existence of stable sheaves. After the exclusion of the rank one case, these sheaves are generically locally free.
For general surfaces there are existence results by Taubes [23], Gieseker [24], Artamkin [25], Friedmann, J. Li, etc. In particular, bundles with r = 2 and c 1 exist for More generally, W. P. Li and Z.Qin have shown that µ-stable bundles with given r and c 1 exist if where α is a numerical constant depending on the surface D, r, c 1 and the class J. Also, under the assumption c 1 = 0, constant α can not be universal depending on r and c 1 only. It must, in general, depend on J. In particular, on Hirzebruch surfaces, there exists a c 1 such that for any c 2 we can find an ample class J such that M J (r, c 1 , c 2 ) is empty.
A stable bundle W on a hypersurface D in M gives us a point in CH , by using the Grothendieck-Riemann-Roch formula: where i : D ⊂ M denotes the inclusion. Using (2.12) Thus, the general import of the results above is that PCH (J) and PCH contain the points for any effective divisor D and for sufficiently large c 3 .
No such general existence theorems for stable vector bundles on Calabi-Yau threefolds are known. Of course, we know the tangent and cotangent bundles are stable, as are tensor products of these. Two other well known constructions are the monad construction and the spectral cover construction.
As an example, the Chern character for T Q, the tangent bundle of the quintic hypersurface, is where we denote the restriction of the hyperplane class to the quintic by H Q .
The monad construction was used by Maruyama in [9] to prove his asymptotic result.
We follow here the presentation of [27]. To begin with, fix an integral Kaehler class H and a where n and r are some positive integer (r 3) and s i are r linearly independent global sections of O D (nH) which are base point free. This implies that F is a vector bundle on D.
Composing the restriction map r from M to D with α gives the vector bundleẼ(−D) on X asẼ Provided that n is sufficiently large, it was shown in [9] thatẼ(−D) is stable. We are interested in the stable rank r vector bundle E =Ẽ(−J). Its Chern classes were computed in [27].
The spectral cover construction [28,29] applies only for elliptically fibered varieties. We consider an elliptically fibered Calabi-Yau threefold π : M → B, and construct a rank r bundle V with c 1 = 0, stable respect to some ample class of the form The second Chern class of the the tangent bundle of these threefolds and the second Chern class of these bundles are given by where σ and F denote the zero section and fiber class respectively, η some ample class on B, and m M and m V are positive integers determined by the choice of threefold and the choice of spectral cover. More specifically, the class of the spectral cover is given by [C] = rσ + π * η, with stability ensured for irreducible spectral covers. The existence of such a cover can be guaranteed by imposing the following numerical constraints These results give us a supply of known elements of CH .

Necessary conditions
We now consider necessary conditions on CH . First, we repeat the Bogomolov bound for a µ-stable bundle with respect to J, For Calabi-Yau threefolds this seems to be sharp. On surfaces, evidence so far is consistent with a slightly stronger bound. We postpone the discussion to Appendix A which will fit well with the physics arguments below.
Bounds on the higher Chern classes are discussed in [11]. For example, given any polynomial P (c) in the Chern classes, there exist computable polynomials P L and P H in r, c 1 , ∆ and invariants of M, such that for all semistable reflexive sheaves with fixed r, c 1 J n−1 and 0 ≤ c 2 J n−2 ≤ ∆.
In terms of the invariants Eq. (2.4), the general bound Eq. (2.21) becomes in terms of computable polynomials W i (which again depend on invariants of M).

Mirror symmetry and the attractor mechanism
In this section, which is not strictly necessary for the rest of the discussion, we explain the relation between the IIa string theory in which our problem is naturally formulated, and the mirror IIb string theory. This is useful in part because some of the definitions are more natural in the IIb picture, and in part to explain the string theory corrections.
Thus, we consider the IIb superstring theory [41], compactified onM × R 3,1 . Later we will takeM to be the mirror of M, but for now the discussion is general. Such a compactification leads to a four dimensional N = 2 supergravity theory containing various fields, in particular b 3 /2 abelian gauge fields (thus, copies of Maxwell's theory). These arise because the IIb supergravity theory contains a four-form generalized gauge potential C (4) , satisfying the self-duality relation (up to non-linear terms we can ignore), where * is the ten-dimensional Hodge star. Then, given a class Σ i ∈ H 3 (M, Z), the integrals natural origin for such a BPS state in IIB string theory is a Dirichlet 3-brane. It was shown in [42] that BPS D3-branes can be represented in the large volume limit ofM by a special Lagrangian three-cycle γ L .
As outlined above, we will attempt to decide about the existence of BPS states in the IIB super-gravity approximation of IIB string theory. As shown in [12], equations which were first solved in large volume limit by [15] 3.1 Review of closed string mirror symmetry We generally follow the reviews [16,30]. Now we can introduce the period vectors (3.10) The coordinates X I , I = 1, . . . , h 2,1 (M) + 1 are called projective coordinates on M(M). In addition, it can be shown that there exists a so-called pre-potential F , a homogenous function of degree two in X I such that F I = ∂F (X) ∂X I . It follows from 3.2 that the the holomorphic threeform Ω can be written as The large complex structure limit is a point in M that will be determined by a vanishing These coordinates allow the definition of an in-homogenous pre-potential F (t) = F (X) (X 0 ) 2 which can be explicitly computed [30] and is given to leading order (without instanton corrections) In particular, knowing the in-homogenous pre-potential, allows to express the derivatives of F (X) as (3.14) The special coordinates of the A-model onM can be introduced by b We will always denote the coordinates on the Kaehler moduli space with a tilde, and the coordinates on the complex moduli space without a tilde.

Dirichlet branes and open string mirror symmetry
Our main point here will be to review how the BPS branes we have just discussed are realized as concrete objects in string theory, special Lagrangian 3-cycles onM in the IIb theory, and holomorphic objects on M in IIa theory, where M is the type II mirror ofM . For reviews of this subject, consult [16,30,31] and the upcoming [32].
To define open strings, we introduce Dirichlet branes, which in the first instance are  In general, a B-brane on M which descends from a BPS state in the untwisted theory corresponds to a Π-stable object E * in the derived category D( Coh M). Such an object is a quasi-isomorphism equivalence class of complexes of bundles (E i , d E ), whose charge is In the large volume limit of M, its central charge is which explicitly depends on the Kaehler moduli (B + iJ)(M).
In this work, we will restrict attention to the large volume limit, loosely speaking J >> 1 (in string units) and |F |, |B| << J. In this limit, one expects the Π-stable objects to be µ-stable coherent sheaves which support solutions of the hermitian Yang-Mills equations, possibly with mild singularities. In [2], it is shown that such solutions exist for µ-stable reflexive sheaves, which motivates the precise form of our Conjecture 1.1. Regarding Conjecture 1.2, on surfaces such sheaves are necessarily locally free.
Let us discuss a few of the known stringy corrections to this limit. Some of these can be taken into account by studying a generalization of the hermitian Yang-Mills equations, the MMMS equations [33]. It was shown by Leung [34] that the solvability of these equations follows from a deformed notion of stability. This still leaves further stringy corrections arising from world-sheet instantons. These can be best understood using the mirror symmetry to the IIb string onM . Now applying mirror symmetry directly leads us to the problem of classifying special Lagrangian manifolds, which at present appears more difficult than the original problem. However, an indirect approach is to use mirror symmetry to motivate a deformation of µ-stability for bundles called Π-stability [35,36,38], which is believed to incorporate all the stringy corrections mentioned earlier. A complete description along these lines requires considering not just bundles, but arbitrary objects in the derived category of coherent sheaves. In the one case of a Calabi-Yau threefold which has been completely analyzed at present [39], the total space of the line bundle O P 2 (−3), the set of Π-stable objects is actually simpler in the stringy regime (say near the orbifold point) than at large volume. The set of stable objects varies continuously with [g], so one might be able to start with this simpler description and evolve it up to large volume, providing a new approach to this problem. Doing this would also provide a precise definition of the class of objects which can be used in this limit.
We should also discuss the precise singularities allowed in our sheaves. In general, string theory even allows other objects, such as noncommutative analogs of bundles. We do not know of a complete and precise mathematical definition of the class of allowed objects, other than the implicit definition provided by Π-stability. It appears to be different in the different contexts in which connections appear in the large volume limit (heterotic string, D-branes, and at singularities). In the case at hand of B-type D-branes, it includes the zero size limit of Yang-Mills instantons, and also includes certain "rank one" or noncommutative instantons.
These correspond to torsion-free coherent sheaves, for example the ideal sheaf of a curve.
In this paper, we sidestep such questions by studying the predictions of the attractor conjecture in the large volume limit, and accepting the claim that these correspond to singular solutions of hermitian Yang-Mills. The conjecture itself is more general and it would be interesting to compare it with the classification of these more general objects.
To conclude the subsection, we will give an explicit map between H even (M, Q) and The central charge of the corresponding Lagrangian 3-cycle γ V onM is given by we find for Z γ V at the point s α in the large complex structure limit o¡f moduli space (3.30) It follows from local isomorphism 3.19 that we need to evaluate the central charge Z γ V (s) at To begin with, we compute the in-homogenous pre-potential F (H). We find for the and The integral matrix A αβ has no known topological interpretation and is usually fixed by monodromy transformations. We will consider A as a bilinear map H 2 (M) × H 2 (M) → R an leave it undetermined. Then we find for the inhomogenous prepotential Using ∂H ∂s α = −J α we find and We denote the dual basis of {J α } by {C α } and find for central charge Z γ V (H) We expand both sides of 3.28 in powers of H to derive the map H 2 * (M, Q) → H 3 (M, Q). In particular, we express the vector (q I , p J ) in terms of the Chern classes of V. To begin with consider the term proportional to H 3 . We find We will assume which fixes the normalization. After comparing the terms proportional to H i for i = 2, 1, 0 we find Note that we have computed that map in the neighborhood of the specific point H ∈ M Kaehler (M) ≈ M complex (M). However, since the map is topological, henceforth we will assume it holds everywhere in a neighborhood of the large volume limit of M andM .

Attractor equations
The upshot of the preceding section is that we can phrase the IIa attractor problem in the same terms as the IIb problem, by introducing the following objects in H 2 * (M, C), which spans the subspace of H 2 * mirror to H (3,0) (M), and which is mirror to the three-cycle wrapped by the D3-brane. This neglects all stringy α ′ corrections; we will discuss these later. We will consider Eq. (3.4) in the case for non-vanishing rank and vanishing rank separately.

V is a vector bundle on M
We can solve the zero-form term in Eq. (3.4) by writinḡ with a free parameter ξ. The two-form term is then which is solved by writing which can be simplified to where we used c 2 = −ch 2 and ch 3 = 1 2 c 3 in the case c 1 = 0. These are the desired n + 1 equations for the n + 1 parameters J and ξ. Clearly the main effort in solving them is to find a real two-formH satisfying We choose this normalization ofH to have invariance under the rescaling ch → N ch. Then, we can write J = λH We note that λ and thus J are also invariant under the rescaling ch → N ch.
What are the consistency conditions on such a solution? We define ATT to be the largest set of solutions we might consider, by requiring the classH to lie in the closure of the Kähler cone. This implies that for all J. Since for Calabi-Yau threefolds c 2 (M) · J 0, any point in ATT will satisfy the Bogomolov bound, for any J.
In deciding which points in ATT correspond to stable bundles, we must discuss the stringy corrections to the problem. First, there are corrections to Eq. (4.1). These are in principle entirely determined by mirror symmmetry and the considerations of the previous section; we haveΩ The form ω 3 is a six-form whose integral on M is α ′3 , i.e. 1 in "string units." Similarly, the exponentially small terms are functions of volumes of cycles measured in string units.
Since the original problem of solving the Hermitian Yang-Mills equations or finding µstable bundles did not require defining the string unit of length, it would appear that these corrections are not directly relevant, and we will neglect them in the following. However, before proceeding, we should explain what role they play in the full problem arising in string theory. First, there are α ′ corrections to the HYM equations and the stability condition as well, as discussed at length in [16]. Second, even if we start at large volume (meaning J >> 1 in string units, so that world-sheet instanton corrections are negligible), the attractor point might be at small volume. We would then expect the BPS state to be stable at the attractor point, where stringy corrections might be important.
For both reasons, a full treatment of the problem would involve comparing the exact attractor points to the set of Π-stable objects. However, since (as we will see later) there are unresolved issues even before we reach this point, we will leave this for future work.
Given c 2 andH, the remaining step is to vary ξ and see what values of c 3 can be attained.
Since the quantity ξ/(r 2 + ξ 2 ) 1/2 takes all values between −1 and 1, we conclude that all c 3 can be attained which satisfy Note that this bound implies that we find (4.13) and 14) The zero and two form term in Eq. (3.4) are solved bȳ where ξ is a free real parameter. To solve the four form term we have to find a B such that This equation simply implies that c 1 is given as the restriction of a class in H 2 . We will denote this lift byc 1 . The strong Lefschetz theorem implies that the intersection is an isomorphism. We can invert Eq. (4.16) and find Finially, the six form part of Eq. (4.4) implies and therefore at fixed r and H, for large n our bound will be violated. Thus, we can only claim that it is a sufficient condition, which is all that our arguments guaranteed so far.

Bound states
What are the physical states corresponding to stable bundles violating our bounds? So far, we have only considered BPS states corresponding to spherical symmetric black hole solutions in four dimensions. A more general class of solutions is the bound states of spherical symmetric solutions, as discussed by Denef [17]. Such a solution will have a charge which is the sum of charges of two or more constituents, each of which should lie in ATT . Because of these solutions, the condition Z(γ, B + iJ) = 0 does not rule out the existence of a BPS state of charge γ, as we now explain.
Not all sums of elements of ATT correspond to multi-center solutions. As shown in [17,18,19,20,21], a multi-center solution corresponds to a "split attractor" flow. Mathematically, this is an embedding of an oriented tree into moduli space, such that the edges are gradient flows, the endpoints are attractor points, and the vertices are additive decompositions of the charge vector, satisfying the following two consistency conditions. First, the intersection product Eq. The start point (call it t 0 ) of the split attractor flow (as for other attractor flows) is physically the value of the moduli "in the asymptotic region," so mathematically the existence or non-existence of a flow implies stability at t 0 , either µ-stability for t 0 >> γ (the large volume limit), or Π-stability in general.
In general, the analysis of the existence of these flows is rather intricate, and in the In general, this is a statement about Π-stability. It would be of direct interest for our original goal of characterizing µ-stable bundles, if the zero was attained in the large volume limit, J >> |ch(V )| (taking say the l 1 norm). However, it is easy to see that this will not happen, using the fact that large volume formula Eq. (1.13) is polynomial in ch(V ), and elementary bounds on the locations of zeroes of polynomials. Still, it is interesting for example that µ-stable bundles which violate Eq. (4.9) will have lines of marginal stability at some J ∼ chV , analogous to those for high degree hypersurfaces [36,17].
To further illustrate the nature of the split attractor condition, let us now consider a simpler set of necessary conditions for the existence of a split attractor flow.
One simple criterion for when two states of charges γ ′ and γ ′′ in H 3 (M ) and central charge Z ′ and Z ′′ can form a bound state is that This condition follows from the analysis of the gradient flow for Z ′ , Z ′′ and Z. To see this, if we assume that the gradient flow for a BPS state of charge γ (parameterized by the time τ ∈ (0, ∞)) crosses a line of variation of stability and decomposes into two states of charge γ ′ and γ ′′ respectively, then the time τ vs can be explicitly computed by hence 5.2 follows.
One expects that on the bundle side, such a bound state will correspond to an extension of bundles, which can exist if Ext 1 (V ′′ , V ′ ) = 0 and µ(V ′ ) < µ(V ′′ ). For BPS states, there is not such a clear distinction between positive and negative rank, and one might also find bound states which correspond to sub or quotient bundles, These will exist if there is a non-trivial map α ∈ Hom(V ′ , V " ) with vanishing kernel or cokernel.
On the bundle side, these are not easy conditions to check in general, and one cannot expect to find a simple sufficient condition like Eq. (5.2). However, the hope would again be that in an asymptotic limit of large Chern character, a simple condition emerges. Thus, let us use the correspondence between special Lagrangian three-cycles and stable bundles Eq.
We find for two bundles V ′ and V ′′ with Chern character ch ′ and ch that for large Chern character this condition becomes generic, it is reasonable to expect the closure of ATT under this construction to become a better approximation to CH (J).
Thus, we define the "J-closure" S J of a subset S ⊂ H to be the smallest set containing S and all sums γ ′ + γ ′′ made from pairs satisfying Eq. (5.2).
Let us compare ATT J to CH (J). We begin with a pair of line bundles O M (pJ) and O M (qJ) for some ample J; for p > q one expects to construct subbundles of the form for various m and n. Since ATT and the condition Eq. (5.2) are homogeneous, we will not get a strong constraint on m and n; indeed this pair satisfies Eq. (5.2) for n > m. Thus, let us take m = p and n = q, to get c 1 (E) = 0.
We find that ATT J contains the Chern characters The ratio c 2 3 r/c 3 2 ∼ (p + q) 2 /pq grows without bound, so these certainly include points not in ATT .
On the other hand, they appear consistent with (an assumed homogeneous version of) Maruyama-Langer, as we have bounded |c 3 |r/c 2 2 ∼ (p + q)/pq.
Unfortunately, if we proceed to consider bound states of a pair of these bundles, we find that these can attain arbitrarily large c 3 at fixed c 2 , so they satisfy no interesting bounds at all. Now as we explained, we have only considered a subset of the necessary conditions for a split attractor flow, so there is no contradiction at this point; rather we conclude that a more detailed analysis is required.
Such an analysis appears rather non-trivial and thus let us outline some of the possible outcomes as a guide for future work.
One immediate mathematical question is whether the actual bounds on Chern characters of µ-stable bundles are homogeneous or not. If not, we cannot expect to duplicate the bound by this analysis.
It is not at present clear to us whether the bounds Eq. (2.21) discussed in [11] are homogeneous or not. If they are, then a necessary condition on the third Chern class (for bundles with vanishing first Chern class) could take the form for any ample class J with respect the bundle is stable. The first term is the correction which comes from the term of order α 3 in 4.8, the second term is our usual bound for elements in ATT and the third term (with undetermined constant c) we have included to obtain consistency with known constructions, in particular, with our examples from the monad construction.
To summarize, our attempts to enlarge ATT to include bound states as a candidate description of all of CH remain inconclusive. One possible explanation is that the actual mathematical bound on c 3 is not homogeneous, in such a way that the set PCH simply contains all charges consistent with Bogomolov. Our simplified criterion for bound states reproduces this, leading to a consistent if uninteresting picture. An alternative possibility is that a homogeneous necessary condition exists, such as Eq. (5.11), and that a refined version of the bound state condition would make contact with this. Finally, it is conceivable that, even if the correct mathematical bound is not homogeneous, incorporating higher genus corrections to the attractor conditions (as discussed in [45,46]) which do not respect homogeneity might provide a better description of CH .
Finally, we should note that the definition of ATT given in section 4.1 makes sense in arbitrary dimensions, not just d ≤ 3, and it would be interesting to test the analogous conjecture there. While the physical arguments do not directly apply to this case, many elements of the physical discussion (notably the existence of boundary states and the central charge formulas) do generalize, at least to Calabi-Yau manifolds of arbitrary dimension.
One cannot expect the conjecture to give a very good picture of CH for d ≥ 4; for example we have dim R ATT = 2 + 2b 1,1 < dim CH , but the claim that ATT ⊂ CH still appears reasonable.
valuable discussions, and particularly F. Denef for a critical reading of the manuscript and numerous suggestions.
This work was supported in part by DOE grant DE-FG02-96ER40959.

Appendix A
On solid mathematics grounds, the only known general result is the Bogomolov bound Eq. (1.12). This bound however can be improved if we restrict to certain algebraic surfaces.
First note that stable bundles must be simple, that is In Appendix B we give an example which shows that this improvement cannot be generalized to Calabi-Yau threefolds.
implies the existence of global sections of K, a contradiction. From the defining sequence of K it is clear that H 0 (X, K) vanishes for a generic map β. To see that H 0 (X, ∧ 2 K) = 0, consider h 0 (X, ∧ 2 K) = h 0 (X, K * ⊗ det K) = h 3 (X, K ⊗ det K * ) The vanishing of h 3 (X, K ⊗ det K * ) follows from the fact that O Q (m) is ample for m > 0.