Stable pairs and BPS invariants
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
- 1 -functions
- 2 BPS rationality
- 3 Local definition of BPS invariants
- A Reduced curve classes
- B Stable pairs on Gorenstein curves
- C surfaces
Let be a nonsingular, projective, Calabi-Yau 3-fold
the sheaf is pure,
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 , but the fixed-determinant deformation theory of the associated complex
are conjecturally equal to the reduced DT invariants of . Let
be the generating series. Calculations in the toric Calabi-Yau case can be found in .
Since is Calabi-Yau, the above deformation theory of complexes is self-dual in the sense of . 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 , 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 irreducible
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.
For irreducible, the rational function is invariant under the transformation .
In fact, we prove satisfies the full BPS rationality conjectured in .
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 ,
the Katz-Klemm-Vafa  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.
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,
The Yau-Zaslow formula was proven in the primitive
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.
Let be a nonsingular projective variety over . A nonzero class is effective if is represented by an algebraic curve.
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
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.
The obstruction theory of  can be described as follows. Let
be the projection. There is a universal stable pair ,
over . Let be the associated complex (with in degree 0). Consider the complex
of trace-free Exts, where denotes the relative canonical bundle. In , 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
which is the definition of self-duality in . ∎
For any scheme , Kai Behrend  defines a canonical constructible function
depending only on the local scheme
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 , if is a smooth map of relative dimension , then
If is the Euler characteristic of the fibre of , then
On and ,
we obtain functions
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.
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:
Here, is a sufficiently small analytic neighborhood of . Since -functions depend only on the local scheme structure , 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.
2. BPS rationality
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
For an irreducible class, for all .
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 . ∎
Let be an irreducible class. The invariants satisfy the following identities:
An element of yields an exact sequence
where has 0-dimensional support. We obtain the inequality
If is a line bundle on a nonsingular curve , then Serre duality relates and . More generally, there is a map
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
We stratify by the dimension of ,
where is the locus of sheaves with . There are induced stratifications of and . By , 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
By Proposition 2.2, the generating series
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 .
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.
Arend Bayer  and Yukinoba Toda  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  to equate the invariants to the reduced DT invariants of . For any
we have the obvious exact triangle
Taking the derived dual gives
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  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
and denotes the virtual number of elements of the moduli space of stable objects of the corresponding type. For us, (2.13) predicts
Toda  has gone further with wall crossings for arbitrary (rather than irreducible) stable pairs. Using the work of Joyce , 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  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
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
can be written uniquely as
for integers .
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 .
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 . ∎
In this case the interpretation of the formula (3.3) is clear. For any manifold , the generating function for the numbers is
Therefore , the contribution of to , is
Since is precisely the contribution of to (3.3), we have proved the following.
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:
and so on. Here and below we denote a smooth compact 2-manifold of genus by .
The contribution of the term to is then
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:
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