Stable pairs and BPS invariants

Stable pairs and BPS invariants

Abstract.

We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend’s constructible function approach to the virtual class. For irreducible curve classes, we prove the stable pairs generating function satisfies the strong BPS rationality conjectures.

We define the contribution of each curve to the BPS invariants and show the contributions lie between the geometric genus and arithmetic genus of . Complete formulae are derived for nonsingular and nodal curves.

A discussion of primitive classes on surfaces from the point of view of stable pairs is given in the Appendix via calculations of Kawai-Yoshioka. A proof of the Yau-Zaslow formula for rational curve counts is obtained. A connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.

0. Introduction

Let be a nonsingular, projective, Calabi-Yau 3-fold1. Invariants counting curves in via stable pairs have been defined in [26]. A pair consists of a sheaf on supported in dimension 1 together with a section . A pair is stable if

  1. the sheaf is pure,

  2. the section has 0-dimensional cokernel.

By purity (i), every nonzero subsheaf of has support of dimension 1. As a consequence, the scheme theoretic support of is a Cohen-Macaulay curve. The support of the cokernel (ii) is a finite length subscheme . If the support is nonsingular, then the stable pair is uniquely determined by . However, for general , the subscheme does not determine and .

Discrete invariants of a stable pair include the holomorphic Euler characteristic and the class . The moduli space parameterizes stable pairs satisfying

Pair stability arises naturally as GIT stability in the study of appropriate quotients [20, 26]. In fact, GIT stability is found there to be equivalent to semi-stability. The moduli space is a therefore a projective scheme.

To define invariants, we use a virtual cycle. The usual deformation theory of pairs is not perfect in the sense of [4], but the fixed-determinant deformation theory of the associated complex

(0.1)

is shown in [26, 13] to define a perfect obstruction theory for of virtual dimension zero. A virtual cycle is then obtained by [4, 22]. The resulting invariants

are conjecturally equal to the reduced DT invariants of [23]. Let

be the generating series. Calculations in the toric Calabi-Yau case can be found in [27].

Since is Calabi-Yau, the above deformation theory of complexes is self-dual in the sense of [3]. Heuristically, may be viewed locally as the critical locus of a function. The virtual dimension of is 0 and, on the nonsingular locus of the moduli space, the obstruction sheaf is the cotangent bundle. Therefore if is everywhere nonsingular then

where denotes the topological Euler characteristic.

If singularities are present, certainly differs from the (signed) Euler characteristic. By Behrend’s results [3], there exists an integer-valued constructible function over any scheme with the property that if the scheme is proper and admits a self-dual obstruction theory then the length of its virtual cycle equals its -weighted Euler characteristic. Therefore

At nonsingular points,

but at singularities is a more complicated function. The weighted Euler characteristic of is a deformation invariant.

Behrend’s theory applied to allows us to use topological Euler characteristics and cut-and-paste techniques. We require new technical results comparing the value of Behrend’s function at a pair to the value at the sheaf , see Theorem 4 of Section 1. The arguments turn out to be remarkably simple when is an irreducible2 class. We prove the following result in Section 2.

Theorem 1.

For irreducible, is the Laurent series expansion of a rational function in .

Serre duality relates a line bundle on a nonsingular curve to . Since

Serre duality relates the geometry of to . The compatibility of Serre duality with proven in Section 2 yields a more subtle result.

Theorem 2.

For irreducible, the rational function is invariant under the transformation .

In fact, we prove satisfies the full BPS rationality conjectured in [26].

Theorem 3.

For irreducible,

(0.2)

where the are integers and is the maximal arithmetic genus of a curve in class .

We obtain a deformation invariant definition of the BPS counts of Gopakumar-Vafa [7, 8] for irreducible classes . In Section 3, we give a local definition of these BPS invariants for irreducible curve classes. We define constructible functions over the space of curves in with respect to which the weighted Euler characteristics yields the BPS numbers. We prove the functions are nonzero on only in genus between the geometric and arithmetic genera of . Complete evaluations of the functions are obtained for nonsingular and nodal curves.

In Appendix A, we sketch the extension of Theorems 1 and 2 to reduced curve classes which are not necessarily irreducible. We also explain what is needed to show the vanishing of BPS counts in negative genus in the reduced case.

Interesting examples of irreducible and reduced classes occur on surfaces. If is Gorenstein, the stable pairs with support are proven in Appendix B to correspond bijectively to finite length subschemes . The moduli spaces of stable pairs on a surface are then shown to be isomorphic to relative Hilbert schemes.

In Appendix C, the beautiful theory of primitive classes on surfaces is considered. By results of Kawai-Yoshioka [17], the Katz-Klemm-Vafa [16] formula for BPS state counts is obtained for the theory of stable pairs. The corresponding calculations in Gromov-Witten theory have not yet been completed.3

Let be the number of rational curves of fixed primitive class with self-intersection on a surface. Using the genus 0 BPS counts together with the local BPS theory of Section 3, a new proof of the Yau-Zaslow formula,

is obtained.

The Yau-Zaslow formula was proven in the primitive4 case by Bryan-Leung [5] via Gromov-Witten theory. Our proof is very close in spirit to the original sheaf-theoretic motivations for the formula [34]. In particular, our argument via stable pairs and BPS counts is parallel to Beauville’s proof using compactified Jacobians and Euler characteristics [2].

Acknowledgements

We thank J. Bryan and D. Maulik for many conversations related to stable pairs and A. Marian for pointing out the basic connection to the work of Kawai-Yoshioka. We are grateful to S. Kleiman for advice on Jacobians of singular curves, and Hua-Liang Chang for a careful reading of the manuscript.

R.P. was partially supported by NSF grant DMS-0500187 and a Packard foundation fellowship. R.T. was partially supported by a Royal Society University Research Fellowship. He also thanks the Leverhulme Trust and Columbia University for a visit to New York in the spring of 2007 when the project was started. Many of the results presented here were found during a visit to Lisbon in the summer of 2007.

1. -functions

Let be a nonsingular projective variety over . A nonzero class is effective if is represented by an algebraic curve.

Definition 1.1.

An effective class is

irreducible if there is no decomposition into nonzero effective classes ,

primitive if is not a positive integer multiple of an effective class,

reduced if in every decomposition into effective classes, all of the are primitive.

For example, classes of minimal degree measured against any ample class are irreducible. Any primitive class on a surface is irreducible on a generic deformation of for which is of type .

Let be a Calabi-Yau 3-fold. If is a a stable pair of irreducible class , is a stable sheaf since all quotient sheaves have 0-dimensional support.5 There is therefore a map

(1.2)

from the moduli space of stable pairs to the moduli space of stable pure sheaves of Hilbert polynomial

Moreover, the fibre of (1.2) over a point is . By the irreducibility of , the cokernel of any nonzero section is -dimensional and is a stable pair.

Since is Calabi-Yau, both [26] and [31] have self-dual obstruction theories. We can therefore apply the results of [3].

Lemma 1.3.

The obstruction theory of obtained from fixed determinant deformations in the derived category [26] is self-dual in the sense of Behrend [3].

Proof.

The obstruction theory of [26] can be described as follows. Let

be the projection. There is a universal stable pair [26],

over . Let be the associated complex (with in degree 0). Consider the complex

(1.4)

of trace-free Exts, where denotes the relative canonical bundle. In [26], the complex (1.4) is shown to be quasi-isomorphic to a 2-term complex of locally free sheaves over , with a canonical morphism

to the cotangent complex of . The morphism is obstruction theory for : the induced maps on and are isomorphisms and surjections respectively.

For Calabi-Yau, is trivial. Therefore, by relative Serre duality for , we obtain a quasi-isomorphism

Thus

which is the definition of self-duality in [3]. ∎

For any scheme , Kai Behrend [3] defines a canonical constructible function

depending only on the local scheme structure6. If is compact and equipped with a self-dual obstruction theory, then

where the right side is the weighted Euler characteristic

and is the usual topological Euler characteristic.

If is nonsingular, then is the constant function and

More generally, by Proposition 1.5(i) of [3], if is a smooth map of relative dimension , then

(1.5)

If is the Euler characteristic of the fibre of , then

On and , we obtain functions and .7 The invariants

are the weighted Euler characteristics.

The following property holds even though the map (1.2) may be neither smooth nor surjective. The result underpins the whole paper.

Theorem 4.

.

Proof.

We work locally around one point of . By the irreducibility of , the Cohen-Macaulay support of is reduced and irreducible. Hence, is generically nonsingular. There exists a local smooth divisor which intersects (and all nearby in the same homology class) transversally in a single point. We may also assume to be disjoint from the zeros of .

Let . Tensoring with and the canonical section yields a map of analytic open sets:

(1.6)

Here, is a sufficiently small analytic neighborhood of . Since -functions depend only on the local scheme structure [3], the bottom isomorphism makes the -functions of the two sheaf moduli spaces locally the same. We call them . The open sets

contain and respectively. For sufficiently large, is a smooth -bundle.

By making smaller if necessary, the map admits a left inverse

given by forgetting about the points close to . The map is locally smooth with fibre the th symmetric product of an open subset of a nonsingular curve.

We calculate the -function of in two different ways round the commutative diagram (1.6), using (1.5) applied to the two smooth maps and . The two resulting expressions are

Pulling back to gives

Multiplying by gives the result. ∎

2. BPS rationality

2.1. Results

Let be a Calabi-Yau 3-fold and an irreducible class. Let be the maximal arithmetic genus of a curve in the class . Following the notation of Section 1, let

denote the invariants of [26, 31].

Proposition 2.1.

For an irreducible class, for all .

Proof.

Let be a stable sheaf determining a moduli point of . Let be the support of . As in the proof of Theorem 4, let be a transverse divisor meeting in 1 point. Let be the analytic open set of sheaves supported on curves with a single transverse intersection with . Tensoring with multiples of makes isomorphic to a corresponding open set in each .

If is covered by finitely many open sets of the above form, the corresponding open sets cover . By construction, the intersections

are isomorphic to the corresponding intersections

Calculating the weighted Euler characteristics of the spaces and as a sum of weighted Euler characteristics of the (minus the weighted Euler characteristics of their intersections, plus the triple intersections and so on), we find . ∎

Proposition 2.2.

Let be an irreducible class. The invariants satisfy the following identities:

(2.3)
(2.4)
(2.5)
Proof.

An element of yields an exact sequence

where has 0-dimensional support. We obtain the inequality

Therefore is empty for , which implies (2.5). We verify (2.3) and (2.4) simultaneously by proving (2.4) for all .

If is a line bundle on a nonsingular curve , then Serre duality relates and . More generally, there is a map

(2.6)

Since is pure, has homological dimension 2 [12, Proposition 1.1.10] so . Similarly because is supported in codimension 2. Therefore

which has the same Chern classes as elements of .

Pick a 3-term locally free resolution of ,

Applying gives a 3-term locally free resolution

of . Therefore by [12, Proposition 1.1.10] is a pure sheaf. By the irreducibility assumption, is stable and indeed defines an element of .

The map (2.6) is an involution and hence yields an isomorphism

We may therefore consider the projections and (1.2) to fibre and over the same space . We have

by Serre duality on . The fibres of and over are therefore

(2.7)

respectively.

We stratify by the dimension of ,

where is the locus of sheaves with . There are induced stratifications of and . By [3], we may calculate the invariants via these stratification as

with the last equality following from Theorem 4. The -function is the same constant on the fibres of both fibrations .

By (2.7), over , is a -bundle and is a -bundle. These fibres have Euler characteristics and respectively. We find

and

Subtracting gives

By Proposition 2.2, the generating series

(2.8)

is the Laurent expansion of rational function in , completing the proof of Theorem 1. However, a stronger statement can be made. Any Laurent series such as (2.8) can be written as

(2.9)

where the sum is over all and only finitely many terms with are nonzero; see [26]. Moreover the integrality of the coefficients of (2.8) is equivalent to the integrality of the .

The conditions (2.32.5) easily imply the vanishing of for and . Therefore, by Proposition 2.2, can be written uniquely in the BPS form

(2.10)

for integers which vanish for and for greater than the largest genus of a holomorphic curve in the class . Since (2.10) is invariant under , Theorems 2 and 3 are proven.

2.2. Remarks

From formula (2.10), we find the genus BPS invariant equals is the DT invariant of sheaves in agreement with the proposal of S. Katz [15]. In fact, Katz expects

to hold in much greater generality.

Our sheaf theoretic definition of BPS invariants (2.10) in the irreducible case is the first rigorous and manifestly deformation-invariant approach. Other papers on the subject [10, 30, 32] have defined BPS invariants following the original perspective of [7, 8] using -actions on sophisticated cohomology theories, but have been unable to incorporate the virtual class. These definitions are therefore unlikely to be deformation invariant. Our definition is rather simpler, and more in line with the viewpoint of [16].

It should be possible to extend our results to the Fano case for any class . After imposing the requisite number of incidence conditions to cut the virtual dimension to 0, the Fano case behaves like the Calabi-Yau case for irreducible , as all other invariants vanish. However, at present, the analogue of is missing in the Fano case.

2.3. Wall-crossing

Arend Bayer [1] and Yukinoba Toda [33] have made the beautiful observation that (2.4) should be seen as a wall-crossing formula. In fact, the wall-crossing is much simpler than the wall-crossing conjectured in [26] to equate the invariants to the reduced DT invariants of [23]. For any

we have the obvious exact triangle

(2.11)

Taking the derived dual gives

(2.12)

where is the sheaf dual to under the duality (2.6).

Start with a stability condition for which the complexes and the sheaves are stable. In particular, the phase of should be less than that of due to the exact triangle (2.11). Now pass through a codimension 1 wall in the space of stability conditions so that the phase of crosses that of . The extensions (2.11) become unstable, while extensions in the opposite direction (2.12) become stable. Therefore, on the other side of the wall, the stable objects are the derived duals of the complexes made out of stable pairs in .

Ideally, wall-crossing should be studied with Bridgeland stability conditions. However, at present, their existence is conjectural. If instead we use Bayer’s polynomial stability conditions or Toda’s limit stability conditions, then the analysis can be made precise. These stability conditions have been constructed, and the stable objects are as claimed above [1, 33].

Since the pieces occurring in the complexes are also stable in these stability conditions, Joyce’s conjectural wall-crossing formula [14] takes a very simple form. We count only complexes of trivial determinant throughout. The invariant counting the stable objects on one side of the wall should differ from those on the other side by

(2.13)

where

and denotes the virtual number of elements of the moduli space of stable objects of the corresponding type. For us, (2.13) predicts

(2.14)

in precise agreement with (2.4). Perhaps (2.14) is the first nontrivial example of a wall-crossing formula in the derived category that can be rigorously proved.

Toda [33] has gone further with wall crossings for arbitrary (rather than irreducible) stable pairs. Using the work of Joyce [14], he proves analogues of Theorems 1 and 2 for the Euler characteristics of the moduli spaces of stable pairs. Once Behrend’s function and the identities of Kontsevich-Soibelman about the value on extensions [19] have been incorporated into Joyce’s work, Theorems 1 and 2 for all classes on Calabi-Yau 3-folds should be obtained.

3. Local definition of BPS invariants

3.1. Fixed curve

Let be a Calabi-Yau 3-fold. Throughout this Section we fix a Cohen-Macaulay curve in the irreducible class of arithmetic genus . The curve is reduced and irreducible. Let

denote the loci of stable pairs and pure sheaves supported on . Define localised invariants8 by

(3.1)

In Proposition 2.2, we computed the weighted Euler characteristics of the spaces using the map (1.2) to . We can instead restrict attention to the loci , the inverse images of . The same proof applies, since is invariant under tensoring by line bundles and under the duality (2.6). We therefore obtain the same identities for localised invariants:

where . Thus, the generating series

(3.2)

can be written uniquely as

(3.3)

for integers .

3.2. Chow

Let denote the variety of 1-dimensional cycles in the class . Since is irreducible, the cycles have no multiplicities. In fact, parameterises Cohen-Macaulay curves in class .

The spaces and map to with fibres and respectively. We may calculate weighted Euler characteristics of and as weighted Euler characteristics of , with weight function the weighted Euler characteristics of the fibres. More precisely, the integers (3.1) define constructible functions

whose weighted Euler characteristics are the integers . Similarly, (3.2) defines a -valued constructible function on with weighted Euler characteristic .

Therefore the (3.3) define constructible functions

such that the BPS invariants of (2.10) are the weighted Euler characteristics

We call the contribution of to .

Since these definitions hide behind a lot of formulae, their exact meaning is rather opaque. We would like to be able to compute the contributions directly, without computing all of the stable pairs invariants . In fact, the invariants are the more fundamental invariants, from which all others (GW, DT, stable pairs) should follow. Naively, we expect [7, 8, 16] that within the class , the invariant counts curves of geometric genus “in ”. Here “in” is to be interpreted loosely, including, as we discover below, maps which are merely generically embeddings. In particular, a nonsingular curve

of genus should contribute only to , while a reduced irreducible curve of arithmetic genus and geometric genus should contribute to at most .

3.3. Nonsingular curves

The BPS contributions of a nonsingular curve are easy to compute. First, we need to understand the local deformation theory of the pairs spaces about the locus of pairs supported on . The answer turns out to be very simple with all of the -functions of these spaces being equal, up to sign, to the same constant . Here, as before, is the -function of the moduli space

of sheaves of Hilbert polynomial .

Lemma 3.4.

is constant.

Proof.

We follow the proof of Theorem 4. As there, in a neighbourhood of the locus

of sheaves supported on , is isomorphic to all other such moduli spaces

where is a divisor on of degree . We use the same argument as before: extend to a local divisor in (using the nonsingularity of ) and map

taking to . Thus

and the -function is identically constant over the loci of sheaves supported on , since is arbitrary. By Theorem 4, the -functions of all of the moduli spaces of pairs take the constant value on restriction to . ∎

The space of pairs supported on is the th symmetric product of (a proof of a more general fact is given in Proposition B.5 below). By Lemma 3.4, the local stable pairs invariants are

In this case the interpretation of the formula (3.3) is clear. For any manifold , the generating function for the numbers is

(3.5)

Therefore , the contribution of to , is

Since is precisely the contribution of to (3.3), we have proved the following.

Proposition 3.6.

A nonsingular curve of genus contributes

to . And for .

3.4. Singular curves: discussion

For smooth curves, the geometry of the formulae (3.3) is very simple. Remarkably, the BPS formalism makes sense in the singular case also. To start, we expand the formulae (3.3) out and read off the inductively:

(3.7)

and so on. Here and below we denote a smooth compact 2-manifold of genus by .

The formulae (3.4) tell us, inductively, what contributes to each BPS number. The moduli space consists of the single point . By Theorem 4, , so by (3.4),

The contribution of the term to is then

(3.8)

If is nonsingular, (3.8) is precisely the contribution of the space of pairs supported on , but for singular is the more complicated weighted Euler characteristic

We define to be the discrepancy between these two Euler characteristics:

(3.9)

For example, consider a curve with 1 node for which the moduli space of sheaves is nonsingular in a neighbourhood of . Let denote the sign . Then , , and so

We proceed inductively by viewing as contributing nonsingular curves of genus and nonsingular curves of genus . These genus and curves contribute

to , as in (3.4). The discrepancy

is what we define to be the number of genus curves supported on .

These formulae quickly become unmanageable, which is why we use the more concise generating functions (3.3), to which they are equivalent. A number of miraculous cancellations of Euler characteristics and -functions must occur for a singular curve of geometric genus to have for . We will obtain these cancellations from an interplay between Serre duality and Theorem 4.

3.5. Singular curves: results

Let