YUNFENG JIANG AND JIAN ZHOU

Transcription

1 COUNTING INVARIANTS FOR O O( 2)-QUIVERS YUNFENG JIANG AND JIAN ZHOU Abstract. We consider the moduli space of quiver-representations for the O O( 2)-quiver. The derived category of such quiver representations is equivalent to the derived category of coherent sheaves of local Calabi-Yau threefold Y with P as the exceptional set and normal bundle O O( 2). Using similar idea of Nakajima and Nagao, we classify the chambers for the Donaldson-Thomas (DT) invariants, Pandharipande-Thomas (PT) invariants and Szendroi noncommutative Donaldson-Thomas (NCDT) invariants in the space of stability conditions. We write down the generating functions for all of these invariants and verify the GW/DT/PT/NCDT-correspondence. Contents. Introduction 2 Acknowledgements Quiver for local O O( 2) Quiver for local O O( 2) Perverse coherent system Stability conditions and wall-crossings θ-stability conditions The walls Counting invariants Wall-crossing formula DT/PT/NCDT-correspondence Comparison to Joyce generalized DT-invariants for quivers GW/DT/NCDT/PT-correspondence The GW/DT-correspondence. 6 References 7 7

2 2 YUNFENG JIANG AND JIAN ZHOU. Introduction Let Y be a smooth Calabi-Yau threefold. There are several curve counting invariants on Y. Gromov-Witten (GW) invariants are virtual count of stable maps from curves to the threefold Y, and Donaldson-Thomas (DT) invariants are virtual count of stable sheaves on Y. The famous MNOP [MNOP] conjecture states that these two invariants are equivalent on the partition functions after change of variables. Recently Pandharipande-Thomas [PT] introduced Pandharipande- Thomas stable pair (PT) invariants as curve counting invariants. A stable pair is an object in the derived category of coherent sheaves on Y, hence PT invariants are virtual counts of objects in the derived category. The DT/PT-correspondence in [PT] can be taken as a wall-crossing formula inside the space of Bridgeland stability conditions, see [PT], [To],[Ba]. If the Calabi-Yau threefold Y is given by a quiver with superpotential, Szendroi [Sz] defined the so called noncommutative Donaldson-Thomas (NCDT) invariants counting cyclic modules of quiver representations. Zhou [Zh] recently proposed a conjecture that relates GW invariants of the threefold Y to the NCDT invariants of the corresponding quiver, which he called the GW/NCDT duality. In this paper we verify this conjecture for the O O( 2)-quiver. Suppose that the threefold Y is a crepant resolution of X with an isolated singularity. Then from [Lau], the exceptional locus is isomorphic to P, with normal bundle O( ) O( ), O O( 2) or O() O( 3). The threefold Y such that the normal bundle for the exceptional locus P is O( ) O( ), is called the resolved conifold. There exists a conifold quiver (Q, W ) with superpotential such that the derived category of coherent sheaves on Y is equivalent to the derived category of conifold quiver representations. In [Sz], Szendroi computed the partition function for the conifold quiver. Nakajima and Nagao [NN], studied the conifold quiver using perverse coherent system and classified the walls of DT and PT invariants. They proved a wall-crossing formula such that one can write down the generating function for NCDT and PT invariants explicitly. Using the results of [BB], they verified the DT/PT/NCDT correspondence for the conifold quiver. In this paper we consider the case Y such that the normal bundle for the exceptional locus P is O O( 2). The quiver for this threefold is constructed in [AK], which we call O O( 2)-quiver. Using similar ideas as in [NN], [Na], we classify the walls for DT/PT invariants for Y and the NCDT invariants for the O O( 2)-quiver (Q, W ). Using localization method we also compute the Gromov-Witten partition function of the threefold Y and prove the GW/DTcorrespondence of [MNOP] for Y by assuming the DT/PT-correspondence of [PT], which is proved by Toda, Bridgeland [Br],[To] modulo inserting Behrend functions. Thus we verify the GW/NCDT duality proposed by Zhou [Zh]. The outline of the paper is as follows. In Section 2 we introduce the quiver for threefold Y such that the exceptional locus P has normal bundle O O( 2). We also briefly talk about the perverse coherent system in [NN]. We define the

3 COUNTING INVARIANTS 3 θ-stability in Section 3. We classify the DT/PT chambers in the space of stability conditions and prove a wall-crossing formula. We prove the DT/PT/NCDTcorrespondence by assuming the DT/PT-correspondence, and compare the computations result to Joyce s generalized DT invariants for quiver with superpotential. In Section 4 we prove the GW/DT-correspondence. Acknowledgements. The first author would like to thank Mathematics Department of Tsinghua University for hospitality and wonderful atmosphere for research during a visit in the summer Quiver for local O O( 2). 2.. Quiver for local O O( 2). Let Y be a crepant resolution of an isolated singularity in a three dimensional analytic space. From [Lau], the exceptional set is isomorphic to P and the normal bundle has three cases: O( ) O( ), O O( 2) and O() O( 3). In this paper we only consider the case Y such that the normal bundle of P is O O( 2). This space Y can be obtained by gluing two copies of C 3, with linear coordinates (x, y, y 2 ) and (w, z, z 2 ) by the following rules for change of coordinates: z = x 2 y + xy k 2, z 2 = y 2, w = x. Define a map from Y to C 4 by: v = z 2 = y 2, v 2 = z = x 2 y + xy k 2, v 3 = wz = xy + y k 2, v 4 = w 2 wz k 2 = y. The image is the affine variety X defined by v 2 v 4 v3 2 + v 3v k = 0, or by a change of coordinates: u 2 + u u u 2k 4 = 0. The exceptional set is given by (x, y = 0, y 2 ) and (w, z = 0, z 2 = 0). Its normal bundle is isomorphic to O( ) O( ) P when k =, and to O O( 2) P when k >. For k >, the associated quiver is (cf. [AK] and the references therein): with superpotential function y 0 a a 2 b b 2 x W = ( ) n(n )/2 x n+ ( ) n(n )/2 y n+ xa b + xa 2 b 2 yb a + yb 2 a 2. Let (Q, W ) be the above quiver. Then the relations I of the quiver are given by the partial derivatives of the superpotential function W : () ( ) n(n )/2 (n + )x n a b + a 2 b 2 ; (2) ( ) n(n )/2 (n + )y n b a + b 2 a 2 ; (3) b x yb ; (4) b 2 x yb 2 ; (5) xa a y;

4 4 YUNFENG JIANG AND JIAN ZHOU (6) xa 2 + a 2 y. We denote by A := C[Q]/I the path algebra of the quiver Q with relations I. Then let Coh(A) be the abelian category of coherent A-modules, which is equivalent to the abelian category of finite quiver Q-representations Perverse coherent system. Let f : Y X be the contract morphism, where X = Spec(R) is afine. Let D b (Coh(Y )) be the bounded derived category of coherent sheaves over Y. As in [NN], a perverse coherence sheaf is an object E D b (Coh(Y ) satisfying the following conditions: () H i (E) = 0 unless i=0, -, (2) R f (H 0 (E)) = 0 and R 0 f (H (E)) = 0, (3) Hom(H 0 (E), F ) = 0 for any sheaf F on Y satisfying Rf (F ) = 0 Let Per(Y/X) D b (Coh(Y )) be the full subcategory of all perverse coherent sheaves. Then Per(Y/X) is the core of a t-structure, and hence an abelian category. Let L be an ample line bundle on Y and let P 0 be the vector bundle given by 0 O r Y P 0 L 0. Then P = P 0 O Y is a projective generator. Denote by the O X -algebra f End Y (P) by A. Let D b (Coh(A)) be the derived category of coherent A-modules. Then we have Theorem 2.. There is an equivalence between the two derived categories: D b (Coh(Y )) D b (Coh(A)) given by the functors Rf RHom Y (P, ) and L A (P) such that when restricted to the Coh(Y ) gives an equivalence between Per(Y/X) and Coh(A). Definition 2.2. A perverse coherent system on N is a triple (F, W, s), where F is a perverse coherent sheaf, W is a vector space and s : W C O Y F is a homomorphism. For perverse coherent system (F, C, s), we denote it by (F, s), and (F, 0, 0) by F. A morphism (F, W, s) (F, W, s ) between two perverse coherent systems is given by morphisms of the corresponding perverse sheaves and vector spaces compatible with the homomorphisms s and s. Let Per(Y/X) be the category of perverse coherent systems. Framed quiver. Let Q be the following quiver by adding a vertex into the quiver Q, with same potential W, see the Figure below. y 0 a a 2 b b 2 x Let à := C[ Q]/I be the path algebra of the new quiver Q. Then we have

5 COUNTING INVARIANTS 5 Theorem 2.3. Per(Y/X) = Coh(Ã). 3. Stability conditions and wall-crossings. 3.. θ-stability conditions. Let ζ = (ζ 0, ζ, ζ ) be a triple of real numbers. For a perverse coherent system F = (F, W, s) Per(Y/X), then we have a corresponding Ã-module V = (V 0, V, V ), let θ eζ (Ṽ ) = ζ 0 dim V 0 + ζ dim V + ζ dim V dim V 0 + dim V + dim V. Definition 3.. A Ã-module Ṽ = (V 0, V, V ) Coh(Ã) is θ e ζ -(semi)stable if for any nonzero Ã-submodule Ṽ Ṽ, θ eζ (Ṽ )( ) < θ eζ (Ṽ ). Remark 3.2. () Any coherent Ã-module Ṽ has the unique Harder- Narasimhan filtration: 0 = Ṽ L+ Ṽ L Ṽ Ṽ 0 = Ṽ such that Ṽ l /Ṽ l+ is θ eζ -semistable for l = 0,,, L and θ eζ (Ṽ 0 /Ṽ ) < θ eζ (Ṽ /Ṽ 2 ) < < θ eζ (Ṽ L /Ṽ L+ ). (2) Any coherent θ eζ -semistable Ã-module Ṽ has a Jordan-Hölder filtration: 0 = Ṽ L+ Ṽ L Ṽ Ṽ 0 = Ṽ such that Ṽ l /Ṽ l+ is θ eζ -stable for l = 0,,, L and θ eζ (Ṽ 0 /Ṽ ) = θ eζ (Ṽ /Ṽ 2 ) = = θ eζ (Ṽ L /Ṽ L+ ). Remark 3.3. () As in [NN], let c be a real number, and let ζ = (ζ 0 + c, ζ + c, ζ + c), then we compute θ eζ (Ṽ ) = θ e ζ (Ṽ ) + c. This means that the θ eζ -(semi)stability is equivalent to the θ eζ - (semi)stability. Hence we can normalize θ eζ such that for Ṽ Coh(Ã), θ eζ (Ṽ ) = 0. (2) Let V = (V 0, V ) be a A-module. Then we have a corresponding perverse coherent sheaf F Per(Y/X). Given a pair of real numbers ζ = (ζ 0, ζ ), we define θ ζ -(semi)stability for V or F by the θ eζ -(semi)stability for Ṽ = (V 0, V, 0) or (F, 0, 0) by taking ζ = (ζ 0, ζ, ζ ) for any ζ. Remark 3.4. From the correspondence between Per(Y/X) and Coh(A) under the equivalence D b (Coh(Y )) D b (Coh(A)), we see that for any F Per(Y/X), dim(h 0 (F )) = dim V 0 and dim(h 0 (F P 0 )) = dim V if V = (V 0, V ) is the corresponding A-module in Coh(A).

6 6 YUNFENG JIANG AND JIAN ZHOU From [NN], let ζ = (ζ 0, ζ, ζ ) be a stability parameter, and v = (v 0, v, v ) (Z >0 ) 3. Theorem 3.5. There exist a coarse moduli space M s e ζ (v) (M ss eζ (v)) of θ e ζ - (semi)stable Ã-modules Ṽ with dimension dim(ṽ ) = (v 0, v, v ). Definition 3.6. Let ζ = (ζ 0, ζ ) be a pair of real numbers. An Ã-module Ṽ = (V 0, V, V ) Coh(Ã) is said to be ζ-(semi)stable if it is θ e ζ -(semi)stable for ζ = (ζ 0, ζ, ζ 0 dim(v 0 ) ζ dim(v )). Chambers corresponding to DT/PT. Let ζ im := (, ) and ζ im,± = ( ± ɛ, ) for a sufficiently small ɛ > 0. Then ζ im is on the line ζ 0 + ζ = 0 in R 2 and ζ im,± are on the two sides of this line. We will see that the line ζ 0 + ζ = 0 is a wall for the stability conditions. Since the morphism f : Y X is projective and the fibres of f have dimensions less than 2 and Rf O Y = O X. Then using Definition 3.6 we have: Proposition 3.7. Let Ṽ = (V 0, V, C) be a coherent Ã-module, and correspond to a perverse coherent system (F, s). Suppose that (F, s) is ζ im, -stable, then F is a coherent sheaf and s is surjective. In other words (F, s) is equivalent to an ideal sheaf I Coh(Y/X) from the following exact sequence 0 I O Y F 0. On the other hand, if for (F, s), s is surjective, then (F, s) is ζ im, -stable. Proposition 3.8. Let Ṽ = (V 0, V, C) be a coherent Ã-module, and correspond to a perverse coherent system (F, s). Suppose that (F, s) is ζ im,+ -stable, then F is a pure sheaf of dimension one and the cokernel coker(s) is 0-dimensional. In other words (F, s) is a stable pair in [PT]. On the other hand, if (F, s) is a stable pair, then (F, s) is ζ im,+ -stable. Proof. Propositions 3.7 and 3.8 are from the same proof as in [NN] The walls. A stability parameter ζ = (ζ 0, ζ ) R 2 is said to be generic if ζ- semistability is equivalent to ζ-stability. In this section we classify the nongeneric stability parameters, which we call walls. Let ζ = (ζ 0, ζ ) be a stability parameter. Lemma 3.9. Let W = (W 0, W, C) be a nonzero θ ζ -stable A-module. Then at least one of the following holds: () dim(w 0 ) = dim(w ) = ; (2) a = a 2 = 0, x = y = 0; (3) b = b 2 = 0, x = y = 0. Proof. We use the similar method as in [NN] and the quiver relations (2.) to prove the lemma.

7 COUNTING INVARIANTS 7 First if W 0 = 0 or W = 0, then we easily get b = b 2 = 0, x = y = 0 or a = a 2 = 0, x = y = 0 from the quiver relations. Suppose that W 0 0, W 0. and Set X 0 = ker(x), Y 0 = ker(y) X = Im(x), Y = Im(y). Then from the quiver relations (3), (4), (5), (6) in (2.), the arrows a, a 2, b, b 2 all restrict to the kernels and images of x, y, hence (X 0, Y 0 ) and (X, Y ) are A- submodules. Then θ ζ -stability of W gives: and ζ 0 dim(x 0 ) + ζ dim(y 0 ) 0 ζ 0 dim(x ) + ζ dim(y ) 0 since we can make ζ 0 dim(w 0 ) + ζ dim(w ) = 0. Since dim(x 0 ) + dim(x ) = dim(w 0 ) and dim(y 0 )+dim(y ) = dim(w ), the above inequality should be equality. So x, y are isomorphisms or x = y = 0. Case I: x = y = 0. In this case, the argument is actually very similar to the conifold case as in [NN]. Consider the following and S 0 = ker(a b ), S = Im(a b ) T 0 = ker(b a ), T = Im(b a ). An easy check from the relations gives that (S 0, T 0 ) and (S, T ) are A-submodules. Then θ ζ -stability of W means that: and ζ 0 dim(s 0 ) + ζ dim(t 0 ) 0 ζ 0 dim(s ) + ζ dim(t ) 0. Again since dim(s 0 ) + dim(s ) = dim(w 0 ) and dim(t 0 ) + dim(t ) = dim(w ), the above inequality should be equality. So (S 0, T 0 ) = (0, 0) or (W 0, W ), and so a, b are isomorphisms or a b = 0. Then from the above analysis we can assume () a, b are isomorphisms and dim(w 0 ) = dim(w ); (2) a i b j = 0, b j a i = 0 for every i, j =, 2. For the second case, let ζ 0 0. Consider A-submodule (ker(a ) ker(a 2 ), 0), then the θ ζ -stability implies that ker(a ) ker(a 2 ) = 0. Since a i b j = 0, we get Im(b ), Im(b 2 ) ker(a ) ker(a 2 ) = 0, and hence b = b 2 = 0. If we choose ζ 0 0, then a = a 2 = 0. For the first case, we have ζ 0 + ζ = 0. Let ζ 0 < 0. Consider the four linear maps a b, a 2 b 2, a b 2, a 2 b. Since x = y = 0, using quiver relations (2.) it is easy to check that the four maps commute to each other. Then we take the common nonzero eigenvector w 0 W 0 of a b, a 2 b 2, a b 2, a 2 b, and let V 0 = Cw 0, V = Ca w 0 + Ca 2 w 0.

8 8 YUNFENG JIANG AND JIAN ZHOU Then we check that (V 0, V ) is a coherent A-submodule. Then using θ ζ -stability of W we get (V 0, V ) = (W 0, W ). If ζ 0 = ζ = 0, then we can use the same method to get (V 0, V ) = (W 0, W ). So dim(w 0 ) = dim(w ) =. Case II: x, y are isomorphisms. Consider the following linear transformations x, a b, a 2 b 2, a b 2, a 2 b. Then using quiver relations (2.) we check that the maps commute to each other up to a sign if n is even. The maps a b, a 2 b 2 0 because otherwise the first relation in (2.) violates. Then we take the common nonzero eigenvector w 0 W 0 of x, a b, a 2 b 2, a b 2, a 2 b, and let V 0 = Cw 0, V = Ca w 0 + Ca 2 w 0. Then we check that (V 0, V ) is a coherent A-submodule. The key point is that the map y reduces to V gives a linear map y : V V. This is because of the relation xa = a y and xa 2 = a 2 y. Then θ ζ -stability of W means that ζ 0 dim(v 0 ) + ζ dim(v ) 0. Take ζ 0 < 0, then dim(v ) dim(v 0 ) =, so dim(v ) = and we get (V 0, V ) = (W 0, W ). If ζ 0 = ζ = 0, then we can use the same method to get (V 0, V ) = (W 0, W ). So dim(w 0 ) = dim(w ) =. Suppose that x = y = 0, then the quiver will be the Kronecker quiver Q which contains two vertices, and two arrows connecting the two vertices. From Kac theorem in [Kac], there is an associated root system R in Z 2 to this quiver Q. The positive roots R + are given by: {(m, m), (m, m + ), (m, m ) m Z >0 }. And (m, m) are positive imaginary roots, (m, m + ), (m, m ) are positive real roots. Let R +,rel and R +,im be the sets of positive real roots and positive imaginary roots respectively. Then from Kac theorem we have: Proposition 3.0. For each positive real root (m, m + ) or (m, m ) in R +,rel, there is a unique, up to isomorphism, A-module C with dimension vector (m, m+) or (m, m ). For each positive imaginary root (m, m) in R +,im, there is a moduli space of A-modules C with dimension vector (m, m). Then from Lemma 3.9 and Proposition 3.0 we classify the relationship between θ ζ -(semi)stable modules and positive roots. Remark 3.. In case (), the θ ζ -stable A-modules are given by the one dimensional pair vector spaces (C, C), and the moduli space should be given by Y. In case (2) and (3), for every positive real root (m, m + ) or (m, m ), there exist a unique θ ζ -stable A-module C with dimension vector (m, m+) or (m, m ). For every positive imaginary root (m, m), when m =, there exist a moduli space of θ ζ -stable A-modules C with dimension vector (, ). But when m >, there exist a moduli space of strictly θ ζ -semistable A-modules C with dimension vector (m, m).

9 COUNTING INVARIANTS 9 Since for a strictly ζ-semistable Ã-module Ṽ = (V 0, V, V = C), one has the Jordan-Hölder filtration 0 = Ṽ n+ Ṽ Ṽ 0 = Ṽ. Here n since Ṽ is not ζ-stable. Then at least we have one (Ṽ l /Ṽ l+ ) is not zero. Then there exist a nonzero θ eζ -stable Ã-module W such that W = 0 and ζ 0 dim(w 0 ) + ζ dim(w ) = 0. And hence there exist a nonzero θ ζ -stable A- module W such that ζ 0 dim(w 0 )+ζ dim(w ) = 0. Recall the nongeneric stability parameters ζ means that the stability is not equivalent to semistability. Thus we prove: Proposition 3.2. The set of nongeneric stability parameters is given by the union of hyperlines in R 2. W α := {ζ R 2 ζ α = 0, α R + } Remark 3.3. It is easy to see the walls are the same as the walls in the conifold case, classified by Nakajima and Nagao in [NN]. From Propositions 3.7 and 3.8, one can see that the wall mζ 0 + mζ = 0 for the imaginary roots (m, m) is the wall between the Donaldson-Thomas and Pandharipande-Thomas invariants. And for m >, the corresponding A-modules V are strictly θ ζ -semistable. It is interesting to compute the counting invariants on this wall so that one can use the wall-crossing formula in [Joyce], [JS] to prove the PT/DT-correspondence. Of course in this case, one needs the Behrend function [Be] for Artin stacks as defined in [JS]. Following Nakajima and Nagao in [NN], we can write down the generating function for the counting invariants without considering this imaginary wall Counting invariants. Let ζ = (ζ 0, ζ ) R 2 be a stability parameter. Let v = (v 0, v ) (Z >0 ) 2 be a dimension vector and let M s ζ (v)( Mss ζ (v)) be the moduli space of ζ-stable (semi-stable) Ã-modules Ṽ with dimension vector v. Let ζ be a generic stability parameter, then M s ζ (v) admits a symmetric perfect obstruction theory in the sense of [Be] since it is the critical locus of trace of the superpotential function W. Then there is a virtual fundamental class [M s ζ (v)]vir for this moduli space with virtual dimension zero and the counting invariant is defined by # vir M s ζ (v) := deg([ms ζ (v)]vir ). From [Be], there exists an integer value constructible function ν Mζ : M s ζ (v) Z such that the counting invariants are given by the weighted Euler characteristic: # vir M s ζ (v) = n Z n χ(ν (n)) = χ(m s ζ (v), ν M ζ ).

10 0 YUNFENG JIANG AND JIAN ZHOU The generating function is given by () Z ζ (q) = v (Z 0 ) 2 χ(m s ζ (v), ν M ζ )q v. Let w = tr(w ) be the trace of the superpotential function. Then from [Sz], the moduli space M s ζ (v) can be taken as the critical locus of the holomorhic function w. Then from [Be], [BG], ν Mζ (P ) = ( ) v 0 ( χ(f P )) for P M s ζ (v), where F P is the Milnor fiber at the point P. This can be seen as follows. Since the point P M s ζ (v) corresponds to a framed A-module Ṽ, from Corollary in [Sz], we compute that dim(t P M s ζ (v)) = dim(t P M s ζ (v)) = v2 0 + v 2 + 4v 0 v + v 0 v 2 0 v 2 v 0 (mod 2). Then from [Be], ν Mζ (P ) = ( ) v 0 ( χ(f P )) for P M s ζ (v), where F P is the Milnor fiber at the point P. Let I be an index set such that {M s ζ (v) i} i I is a stratification of the moduli space M s ζ (v) satisfying χ(f P ) is constant for P M s ζ (v) i. We denote by χ(f i (w)) for this Euler characteristic. Then we have χ(m s ζ (v), ν M ζ ) = i I ( ) v 0 ( χ(f i (w))) χ(m s ζ (v) i). We may write () as (2) Z ζ (q) = v (Z 0 ) 2 i I ( ) v 0 ( χ(f i (w))) χ(m s ζ (v) i)q v Wall-crossing formula. The set of nongeneric stability parameters are given by the walls in Proposition 3.2. Each component of the generic stability parameters is called a chamber. Inside the same chamber the moduli spaces M ζ (v) are isomorphic. On different chambers, the corresponding invariants of M ζ (v) and M ζ (v) are related by wall-crossing formula. Let α R +,rel be a positive real root. Let ζ o W α be a stability parameter on the wall Q α. Let ζ ± = ζ o ± ɛ. Then the ζ ± lie in two chambers on two sides of the wall W α. Let C be the unique ζ o -stable A-module such that ζ 0 dim(c 0 ) + ζ dim(c ) = 0. Proposition 3.4. ([Na]) If Ṽ is a ζo -stable Ã-module, then ext e A (C, Ṽ ) ext e A (Ṽ, C) = dim(c 0). Lemma 3.5. Ext e A (C, C) = C.

11 Proof. According to the derived equivalence COUNTING INVARIANTS D b (Coh(Y )) D b (Mod(A)), and the restriction to Per(Y/X) gives an correspondence Per(Y/X) Mod(A), every stable A-module corresponds to a perverse sheaf on Y. Since the derived category D b (Coh(Y )) is generated by line bundles supported on P. We can prove the lemma for line bundles. gives Consider C = O P. Then the spectral sequence E p,q 2 = H p (P, O P O P Λ q N P /Y ) Ext p+q Y (O P, O P ) Ext Y (O P, O P ) = H 0 (P, O O( 2)) H (P, O P ) = C. For any line bundle L P, let z : P Y be the zero section a similar method yields Ext Y (z L, z L) = H 0 (P, O O( 2)) H (P, O P ) = C. The result follows. For a positive integer m, let C m be the unique indecomposable A-module which is the m times successive extensions of C by C, i.e. given by the exact sequence 0 C m C m C 0. From Lemma 3.5, one easily have the following result. Proposition 3.6. ([Na]) Let Ṽ = (V, C) be a zeta + -stable Ã-module, then there exists an exact sequence 0 Ṽ Ṽ m (C m) n m 0, where Ṽ is a ζo -stable A-module. The integers n m and the isomorphism class of Ṽ are determined uniquely and satisfy dim(hom(ṽ, C m )) = m n m min(m, m). Moreover, let N m = m m n m, the composition map C Nm Hom(C m, m (C m ) n m ) Ext e A (C, Ṽ ) is injective. Here the first map is induced by inclusions C m C m (m > m). The second map is given by composing the inclusion C C m and Ṽ Ext A e( m (C m ) n m, Ṽ ). On the other hand, let Ṽ = (V, C) be a zeta -stable Ã-module, then there exists an exact sequence 0 m (C m) n m Ṽ Ṽ 0, where Ṽ is a ζo -stable A-module. The integers n m and the isomorphism class of Ṽ are determined uniquely and satisfy dim(hom(c m, Ṽ )) = m n m min(m, m).

12 2 YUNFENG JIANG AND JIAN ZHOU Moreover, let N m = m m n m, the composition map C Nm Hom( m (C m ) n m, C m ) Ext e A (Ṽ, C) is injective. Here the first map is induced by surjections C m C m (m > m). The second map is given by composing the surjection C m C and Ṽ Ext A e(ṽ, m (C m ) n m ). From Lemma 3.5 and Proposition 3.6, we prove the following wall-crossing formula for the generating function of the counting invariants, which generalizes the wall-crossing formula in [NN], where the moduli space under torus action has finite isolated fixed points. For N Z 0, let M ss ζ o (v) N be the subscheme of M ss ζ o (v) consisting of closed points Ṽ such that ext (C, Ṽ ) = N. Let Ms ζ + (v) (nm) be the subscheme of M s ζ + (v) which contains points Ṽ such that Then there is a canonical morphism from the exact sequence hom(ṽ, C m ) = m n m min(m, m). φ : M s ζ + (v ) (nm) M ss ζ o(v), 0 Ṽ Ṽ m (C m ) nm 0, where v = v m mn m dim(c). From [Be], the canonical constructible function ν Mζ + and ν M ζ o satisfy the following formula ν Mζ + = ( )dim(f l) φ ν Mζ o. Let i I be the stratification of M ss ζ o (v) such that χ(m ss ζ o(v), ν M ζ o ) = ( ) v 0 ( χ(f i (w)))χ(m ss ζ o(v) i). i I Then the parity of tangent space of M ss ζ o (v) is ( ) v 0. Similarly the parity of the tangent space of M s ζ + (v ) is ( ) v 0. It is easy to see the the subschemes M s ζ + (v ) (nm),n are defined by the inverse images of the closed subschemes M ss ζ o (v ) N. Then the restriction of φ gives φ : M s ζ + (v ) (nm),n M ss ζ o(v) N whose fiber is the flag variety Fl((N m ), N). Lemma 3.7. Let dim(fl) be the dimension of the flag variety Fl((N m ), N). Then ( ) dim(f l) = ( ) P m mnm dim(c 0). Proof. The map φ is a fibration, let P M s ζ + (v ) (nm),n and φ(p ) M ss ζ o (v) N. Then ( ) dim(t P (M s ζ + (v ) (nm),n )) = ( ) dim(f l) ( ) dim(t φ(p )(M ss ζ o (v) N )).

13 Hence COUNTING INVARIANTS 3 ( ) dim(f l) ( ) v 0 = ( ) v 0. From the equality v 0 = v 0 + m mn m dim(c 0 ) the result follows. The closed subschemes M ss ζ o (v ) N and M s ζ + (v ) (nm),n are similarly defined according to the exact sequence 0 m (C m ) nm Ṽ Ṽ 0. From Proposition 3.4, the following relation is obvious: (3) M ss ζ o(v) N = M ss ζ o(v)n+dim(c 0). We prove a wall-crossing formula, which generalizes the wall-crossing formula in [Na], where the torus action on the moduli space only has isolated fixed points. Theorem 3.8. Z ζ (q) = ( ( q 0 ) dim(c 0) q dim(c ) ) dim C0 Z ζ +(q). Proof. From (2) and Lemma 3.7 we compute Z ζ +(q) = χ(m ζ +(v ), ν Mζ + )qv v = χ(m ζ +(v ) (nm),n, ν Mζ + )qv v,(n m),n = v,(n m),n = v,(n m),n χ(m ζ +(v ) (nm),n, ( ) dim F l φ ν Mζ o )q v ( ) dim F l χ(fl((n m ); N)χ(M ζ o(v) N, ν Mζ o )q v+p mn m dim(c) = v,n (nm)( ) dim F l χ(fl((n m ); N)q P mn m dim(c) χ(m ζ o(v) N, ν Mζ o )q v = ( ( q) dim(c)) N ( ) v 0 ( χ(f i (w)))χ(m ζ o(v) N,i )q v, v,n i I where we use ( q) dim(c) to represent ( q 0 ) dim(c 0) q dim(c ). Using the same arguments we compute that Z ζ (q) = v χ(m ζ (v ), ν Mζ )qv = v,n ( ( q) dim(c)) N ( ) v 0 ( χ(mf i (w)))χ(m ζ o(v) N i )q v. i I

14 4 YUNFENG JIANG AND JIAN ZHOU Then from (3), Z ζ (q) = v χ(m ζ (v ), ν Mζ )qv = ( ( q) dim(c)) N ( ) v 0 ( χ(f i (w)))χ(m ζ o(v) N i )q v v,n i I = ( ( q) dim(c)) N ( ) v 0 ( χ(f i (w)))χ(m ζ o(v ) N dim(c0 ),i)q v v,n i I = ( ( q) dim(c)) N dim(c 0 ) ( ) v 0 ( χ(f i (w)))χ(m ζ o(v ) N,i )q v v,n i I ( ( q) dim(c)) N ( ( q) dim(c)) dim(c 0 ) = v,n ( ) v 0 ( χ(f i (w)))χ(m ζ o(v ) N,i )q v i I = ( ( q) dim C ) dim C 0 Z ζ +(q) = ( ) ( q 0 ) dim(c0) q dim(c dim C0 Z ζ +(q). ) 3.5. DT/PT/NCDT-correspondence. Let I n (Y, d) be the moduli space of rank one ideal sheaves I such that if Z Y is the associated closed subscheme, [Z] = d H 2 (Y, Z) and χ(o Z ) = n. Then we have the Behrend function The Donaldson-Thomas invariant is the weighted Euler characteristic. Let be the generating function. ν I : I n (Y, d) Z. I n,d = χ(i n (Y, d), ν I ) Z DT (Y, q, t) = n,d I n,d q n t d Similarly, let P n (Y, d) be the moduli space of rank one stable pairs O Y F such that if Z Y is the associated closed subscheme, [Z] = d H 2 (Y, Z) and χ(f ) = n. Then we have the Behrend function The Pandharipande-Thomas invariant ν P : P n (Y, d) Z. P n,d = χ(p n (Y, d), ν P ) is the weighted Euler characteristic. Similarly let Z P T (Y, q, t) = n,d P n,d q n t d

15 be the generating function. COUNTING INVARIANTS 5 For a ideal sheaf I or a stable pair O Y F, and the corresponding A-module V with dimension vector v, it is to compute that n = v 0, d = v 0 v. So Z DT (Y, q 0 q, q ) = Z ζ im, (q), Z P T (Y, q 0 q, q ) = Z ζ im,+(q). From the wall-crossing formula in Theorem 3.8. Theorem 3.9. Z NCDT (q) = m ( ( q 0) m q m+ ) Z m DT (Y, q 0 q, q ) and Z P T (q) = m ( ( q 0) m q m ) m Comparison to Joyce generalized DT-invariants for quivers. In the paper [JS], Joyce and Song defined the generalized DT-invaraints in the abelian category of coherent sheaves Coh(Y ). Using generalized DT-invariants the authors studied the BPS state invariants coming from physics. Let Q be a quiver with superpotential, Joyce and Song also defined the generalized DT-invaraints for the abelian category of quiver representations for the quiver Q. Joyce-Song also defined Noncommutative Donaldson-Thomas(NCDT) invariants for the quiver Q, and prove that this invariant is the same as Szendroi s [Sz] noncommutative Donaldson-Thomas invariants. In this section we classify Joyce-Song generalized DT-invariants for the quiver (Q, W ). Let (Q, W ) be the quiver for O O( 2) over P. Let ζ be a stability parameter such that θ ζ (V ) = 0 for A-module V. Let DT v Q(0) Q be the quiver generalized Donaldson-Thomas invariants and DT ˆ v Q(0) Q the quiver BPS invariants for Y. Then one has DT v Q(0) = DT ˆ v/m Q (0). m 2 m v,m Let NDTQ v (0) Q be the quiver noncommutative Donaldson-Thomas invariants, then we have (4) + NDTQ(0)q v v0 0 qv = ( ( q 0 q ) k ) 2k ( ( q 0 ) k q k ) k ( ( q 0 ) k q k+ ) k. v (Z >0) 2 k

16 6 YUNFENG JIANG AND JIAN ZHOU Taking Log to (4) and using Corollary 7.23 in [JS] we get: ( ) v0 v 0 DT v Q(0)q v0 0 qv v (Z >0) ( 2 ) 2k log( ( q 0 q ) k ) k log( ( q 0 ) k q k ) k log( ( q 0 ) k q k+ ) = k = k,l ( ( ) kl kl 2 l 2 qkl 0 q kl l 2 qkl 0 q (k )l l 2 qkl 0 q (k+)l So from [JS], we compute 2 l,l d, v l 2 0 = v = d ; DT v, v Q(0) = l 2 0 = kl, v = (k )l, k, l ;, v l 2 0 = kl, v = (k + )l, k, l ; 0, otherwise. Then ˆ DT v Q(0) = 2, (v 0, v ) = (k, k), k ;, (v 0, v ) = (k, k ), k ;, (v 0, v ) = (k, k + )l, k 0; 0, otherwise. Remark It can be seen that the BPS state invariants DT ˆ v Q(0) are integers, which means that the integrality conjecture in [JS] and [KS] is true in this case. ). 4. GW/DT/NCDT/PT-correspondence. 4.. The GW/DT-correspondence. Let N be a smooth Calabi-Yau threefold. Fixing a curve class d H 2 (Y, Z). Let M g (Y, d) be the moduli stack of genus g stable maps to Y of degree d. There is a perfect obstruction theory on M g (Y, d) and the virtual dimension is zero. Then the Gromov-Witten invariants N g,d := [M g(y,d)] vir. The reduced Gromov-Witten partition function is defined by the following Z GW (Y ) = Z GW (λ, t) = exp N g,d λ 2g 2 t d d>0 g 0 be the generating function. Then from the result of topological vertex in [LLLZ], Proposition 4.. ( N g,d λ 2g 2 = d g 0 (2sin dλ 2 )2 ).

17 COUNTING INVARIANTS 7 Then we compute Z GW (λ, t) = exp N g,d λ 2g 2 t d d>0 g 0 ( ) t d = exp d d>0 d>0 (2sin dλ 2 )2 ( ) t d = exp d e idλ ( e idλ ) 2 = exp ( d>0 m= m d eidmλ t d ) ( ) = exp m( log( te imλ ) = m= m ( ( q)m t) m. So by GW/DT-correspondence, which is proved in [MOOP], we have Z DT = Z DT (q, t) = The equality q = q 0 q and t = q gives m ( ( q)m t) m. Z DT (q 0 q, q ) = m ( ( q 0) m q m ) m. References [AK] P.S. Aspinwall and S. Katz, Computation of superpotentials for D-Branes, Commun. Math. Phys., 264 (2006) , hep-th/ [Ba] A. Bayer, Polynomial Bridgeland stability conditions and the large volume limit, arxiv: [Be] K. Behrend, Donaldson-Thomas invariants via microlocal geometry, to appear in Ann. of Math., math.ag/ [BB] K. Behrend and J. Bryan, Super-rigid Donaldson-Thomas invariants, math.ag/ [BF] K. Behrend and B. Fantechi, Symmetric obstruction theories and Hilbert schemes of points on threefolds, math.ag/ [BG] K. Behrend and E. Getzler, On holomorphic Chern-Simons functional, preprint. [Bri07] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math., 00(2):37-346, arxiv:math.ag/ [Br] T. Bridgeland, Counting invariants and Hall algebras, preprint. [Joyce] D. Joyce, Configurations in abelian categories. iv. Invariants and changing stability conditions, arxiv:math.ag/ [JS] D. Joyce and Y. Song, A theory of generalized Donaldson-Thomas invariants, arxiv: [Kac] V. Kac, Root systems, representations of quivers and invariant theory, Invariant theory (Montecatini, 982), 74-08, Lecture Notes in Math., 996, Springer, Berlin, 983.

18 8 YUNFENG JIANG AND JIAN ZHOU [KS] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arxiv: [LP] M. Levine and R. Pandharipande, Algebraic cobordism revisited, arxiv:math/ [Lau] H. Laufer, On CP as an exceptional set, Ann. Math. Studies., 00, , Princeton University Press. [Li] J. Li, Zero dimensional Donaldson-Thomas invariants of threefolds, arxiv:math/ [LLLZ] J. Li, C.C. Liu, K. Liu and J. Zhou, A mathematical theory of topological vertex,... [MNOP] D. Maulik, N. Nekrasov, A. Okounkov, R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, I, math.ag/ [MNOP2] D. Maulik, N. Nekrasov, A. Okounkov, R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, II, math.ag/ [MOOP] D. Maulik, A. Oblomkov, A. Okounkov, R. Pandharipande, Gromov-Witten and Donaldson-Thomas correspondence for toric varieties, theory, in preparation. [NN] H. Nakajima and K. Nagao, Counting invariants of perverse coherent sheaves and its wallcrossing, arxiv: [Na] K. Nagao, Derived categories of small toric Calabi-Yau 3-folds and curve counting invariants, arxiv: [PT] R. Pandharipande and R. Thomas, Curve counting via stable pairs in the derived category, arxiv:0707:2348. [PT2] R. Pandharipande and R. Thomas, The 3-fold vertex via stable pairs, arxiv:0709:3823. [PT3] R. Pandharipande and R. Thomas, Stable pairs and BPS invariants, arxiv:07:3899. [Pot] J. Le Potier,Systemes coherents et structures de niveau,astérisque, 24, 43, 993. [Pot2] J. Le Potier, Faisceaux semi-stables et syst emes coherents, In Vector bun- dles in algebraic geometry (Durham, 993), vol 208 of LondonMath. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 995. [Sz] B. Szendroi, Non-commutative Donaldson-Thomas theory and the conifold, arxiv: 0705:349. [Tho] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom., 54, , math.ag/9806. [To] Y. Toda, Curve counting theories via stable objects I. DT/PT correspondence, arxiv: [Zh] J. Zhou, Crepant resolutions, quivers and GW/NCDT duality, arxiv Department of Mathematics, University of Utah, 55 S 400 E JWB 233, Salt Lake city, UT 842, USA address: jiangyf@math.utah.edu Department of Mathematics, Tsinghua University, Beijing, China address: zhou@math.tsinghua.edu.cn