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arXiv:0809.2994v3 [math.AG] 4 Dec 2008
Derived categories of small toric Calabi-Yau
3-folds and curve counting invariants
Kentaro Nagao
December 4, 2008
Abstract
We first construct a derived equivalence between a small crepant res-
olution of an affine toric Calabi-Yau 3-fold and a certain quiver with
a superpotential. Under this derived equivalence we establish a wall-
crossing formula for the generating function of the counting invariants of
perverse coherent sheaves. As an application we provide some equations
on Donaldson-Thomas, Pandeharipande-Thomas and Szendroi’s invari-
ants. Finally, we show that moduli spaces associated with a quiver given
by successive mutations are realized as the moduli spaces associated the
original quiver by changing the stability conditions.
Introduction
This is a subsequent paper of [NN]. We study variants of Donaldson-Thomas
(DT in short) invariants on small crepant resolutions of affine toric Calabi-Yau
varieties.
The original Donaldson-Thomas invariants of a Calabi-Yau 3-fold Y are
defined by virtual counting of moduli spaces of ideal sheaves IZof 1-dimensional
closed subschemes Z ⊂ Y ([Tho00], [Beh]). These are conjecturally equivalent to
Gromov-Witten invariants after normalizing the contribution of 0-dimensional
sheaves ([MNOP06]).
A variant has been introduced Pandharipande and Thomas (PT in short)
as virtual counting of moduli spaces of stable coherent systems ([PTa]). They
conjectured these invariants also coincide with DT invariants after suitable nor-
malization and mentioned that the coincidence should be recognized as a wall-
crossing phenomenon. Here, a coherent system is a pair of a coherent sheaf and
a morphism to it from the structure sheaf, which is first introduced by Le Potier
in his study on moduli problems ([LP93]). Note that an ideal sheaf IZ is the
kernel of the canonical surjections from the structure sheaf OY to the structure
sheaf OZ. So in this sense DT invariants also count coherent systems.
On the other hand, a variety sometimes has a derived equivalence with a non-
commutative algebra. A typical example is a noncommutative crepant resolution
of a Calabi-Yau 3-fold introduced by Michel Van den Bergh ([VdB04], [VdB]).
In the case of [VdB04], the Abelian category of modules of the noncommuta-
tive crepant resolution corresponds to the Abelian category of perverse coherent
sheaves in the sense of Tom Bridgeland ([Bri02]). Recently, Balazs Szendroi pro-
posed to study counting invariants of ideals of such noncommutative algebras
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([Sze]). He call these invariants noncommutative Donaldson-Thomas (NCDT in
short) invariants. He originally studied on the conifold, but his definition works
in more general settings ([Youb], [MR]).
Inspired by his work, Hiraku Nakajima and the author introduced perverse
coherent systems (pairs of a perverse coherent sheaf and a morphism to it from
the structure sheaf) and study their moduli spaces and counting invariants
([NN]). This attempt seems successful since
• we can describe explicitly a space of stability parameters with a chamber
structure, and
• at certain chambers, the moduli spaces in DT, PT and NCDT theory are
recovered.
Moreover, in the conifold case, we established the wall-crossing formula for
the generating functions of counting invariants of perverse coherent systems
and provide some equations on DT, PT and NCDT invariants. The chamber
structure and the wall-crossing formula formally look very similar to the counter
parts for moduli spaces of perverse coherent sheaves on the blow-up of a complex
surface studied earlier by Nakajima and Yoshioka [NYa, NYb].
The purpose of this paper is to show the wall-crossing formula (Theorem
2.18) for general small crepant resolutions of toric Calabi-Yau 3-folds. Here
we say a crepant resolutions of affine toric Calabi-Yau 3-fold is small when the
dimensions of the fibers are less than 2. In such cases, the lattice polygon in
R2corresponding to the affine toric Calabi-Yau 3-fold does not have any lattice
points in its interior. Such lattice polygons are classified up to equivalence into
the following two cases:
• trapezoids with heights 1, or
• the right isosceles triangle with with length 2 isosceles edges.
In this paper we study the first case. Our argument works for the second case
as well.
In §1, we construct derived equivalences between small crepant resolutions
of affine toric Calabi-Yau 3-folds and certain quivers with superpotentials. In
§1.1, using toric geometry, we construct tilting vector bundles given by Van den
Bergh ([VdB04]) explicitly. Then, we review Ishii and Ueda’s construction of
crepant resolutions as moduli spaces of representations of certain quivers with
superpotentials ([IU]) in §1.2. In §1.3 we show the tautological vector bundles
on the moduli spaces coincide with the tilting bundles given in §1.1. Using such
moduli theoretic description, we calculate the endomorphism algebras of the
tilting bundles in §1.4.
The argument in §2 is basically parallel to [NN]. In our case, the fiber on
the origin of the affine toric variety is the type A configuration of (−1,−1)-
or (0,−2)-curves. A wall in the space of stability parameters is a hypersurface
which is perpendicular to a root vector of the root system of typeˆA. Stability
parameters in chambers adjacent to the wall correspondingto the imaginary root
realize DT theory and PT theory ([NN, §2]). Note that the story is completely
parallel to that of typeˆA quiver varieties (of rank 1), which are the moduli spaces
of framed representations of typeˆA preprojective algebras ([Nak94], [Nak98],
[Nak01]). Quiver varieties associated with a stability parameter in a chamber
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adjacent to the imaginary wall realize Hilbert schemes of points on the minimal
resolution of the Kleinian singularitie of type A, whose exceptional fiber is the
type A configurations of (−2)-curves ([Nak99], [Kuz]).
Our main result is the wall-crossing formula for the generating functions of
the Euler characteristics of the moduli spaces (Theorem 2.18). The contribution
of a wall depends on the information of self-extensions of stable objects on the
wall. Note that in the conifold case ([NN]) every wall has a single stable object
on it and every stable object has a trivial self-extension. Computations of self-
extensions are done in §2.6.
Note that the sets of torus fixed points on the moduli spaces in DT, PT and
NCDT theory are isolated, and we can show that DT, PT and NCDT invariants
coincide with the Euler characteristics of the moduli spaces. In particular, the
wall-crossing formula provides a product expansion formula of the generating
functions of PT invariants. The indices in this formula are nothing but the BPS
state counts ng,β([GV], [HST01], [Tod]) in the sense of Pandeharipande-Thomas
([PTa, §3]). Although an algorithm to extract Gopakumar-Vafa invariants of
our toric Calabi-Yau 3-folds from the topological vertex expression is known
([IKP]1), the explicit formula in this paper is new as far as the author knows.
In §3, we provide alternative descriptions of the moduli spaces. Given a
quiver with a superpotential A = (Q,ω), we can mutate it at a vertex k to
provide a new quiver with a superpotential µk(A) = (µk(Q),µk(ω)). For a
generic stability parameter ζ, we can associate a sequence k1,...krof vertices
and the moduli space of ζ-stable A-modules is isomorphic to the moduli space
of cyclic modules over the quiver with the potetial µkr◦ ··· ◦ µk1(A). As an
application, we show that for a stability parameter ”between DT and NCDT”
the set of torus fixed points on the moduli space is isolated.
As in [NN], our formula does not cover the wall corresponding to the DT-
PT conjecture. We can provide the wall-crossing formula for this wall applying
Joyce’s formula ([Joy])2. In §4, we make some observations on how the virtual
counting version of the wall-crossing formula would be induced from the per-
spective of the recent work of Kontsevich and Soibelman ([KS]). In fact, the
wall-crossing formula (and hence the invariants) coincides with the Euler char-
acteristic version up to sign. The author learned from Tom Bridgeland that he
and Balazs Szendroi reproved the Young’s product formula ([Youa]) for NCDT
invariants of the conifold. Their idea looks quite similar to our observation.
Acknowledgement
The author is grateful to Hiraku Nakajima for collaborating in the paper [NN]
and for many valuable discussion.
He thanks Kazushi Ueda for patiently teaching his work on brane tilings,
Tom Bridgeland for explaining his work with Balazs Szendroi, Yan Soibelman
for helpful comments on §4.
He also thanks Yoshiyuki Kimura and Michael Wemyss for useful discus-
sions, Akira Ishii, Yukari Ito, Osamu Iyama, Yukiko Konishi, Sergey Mozgovoy
1The author was informed on this reference by Yukiko Konishi.
2Yukinobu Toda informed me that it is possible to prove the (Euler characteristic version
of) DT-PT correspondence conjecture for arbitrary projective Calabi-Yau 3-folds using Joyce’s
formula.
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and Yukinobu Toda for helpful comments. The author is supported by JSPS
Fellowships for Young Scientists No.19-2672.
1Derived equivalences
1.1 tilting generators
Let N0> 0 and N1≥ 0 be integers such that N0≥ N1and set N = N0+ N1.
We set
I = {1,...,N − 1},
ˆI = {0,1,...,N − 1},
˜I =
?1
?
2,3
n +1
2,...,N −1
2
?
,
˜Z =
2
???n ∈ Z
?
.
For l ∈ Z and j ∈˜Z, let l ∈ˆI and j ∈˜I be the elements such that l−l ≡ j−j ≡ 0
modulo N.
We denote by Γ the quadrilateral (or the triangle in the cases N1= 0) in
R2= {(x,y)} with vertices (0,0), (0,1), (N0,0) and (N1,1). Let M∨≃ Z3be
the lattice with basis {x∨,y∨,z∨}, and we identify the plane
{(x,y,1)} ⊂ M∨
with the one where the quadrilateral Γ is. Let M be the dual lattice of M∨.
We denote the cone of Γ in M∨
Rby ∆ and consider the semigroup
R:= M∨⊗ R
S∆= ∆∨∩ M := {u ∈ M | ?u,v? ≥ 0 (∀v ∈ ∆)}.
Let R = RΓ := C[S∆] be the group algebra and X = XΓ := Spce(RΓ) the
3-dimensional affine toric Calabi-Yau variety corresponding to ∆.
Let {x,y,z} ⊂ M be the dual basis. The semigroup is generated by
X := x,
Y := −x − (N0− N1)y + N0z,
Z := y,
W := −y + z,
and they have a unique relation X + Y = N0Z + N1W. So we have
(1)
(2)
(3)
(4)
R ≃ C[X,Y,Z,W]/(XY − ZN0WN1).
A partition σ of Γ is a pair of functions σx:˜I →˜Z and σy:˜I → {0,1} such
that
• σ(i) := (σx(i),σy(i)) gives a permutation of the set
??1
2,0
?
,
?3
2,0
?
,...,
?
N0−1
2,0
?
,
?1
2,1
?
,
?3
2,1
?
,...,
?
N1−1
2,1
??
,
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• if i < j and σy(i) = σy(j) then σx(i) > σx(j).
Giving a partition σ of Γ is equivalent dividing Γ into N-tuples of triangles
{Ti}i∈˜Iwith area 1/2 so that Tihas (σx(i) ± 1/2,σy(i)) as its vertices. Let Γσ
be the corresponding diagram, ∆σ be the fun and fσ: Yσ→ X be the crepant
resolution of X.
We denote by Dε,x(ε = 0,1 and 0 ≤ k ≤ Nε) the divisor of Yσcorresponding
to the lattice point (x,ε) in the diagram Γσ. Note that any torus equivariant
divisor is described as a linear combination of Dε,x’s. For a torus equivariant
divisor D let D(ε,x) denote its coefficient of Dε,x. The support function ψDof
D is the piecewise linear function on |∆σ| such that ψD((x,ε,1)) = −D(ε,x) and
such that ψDis linear on each cone of ∆σ. We sometimes denote the restriction
of ψDon the plane {z = 1} by ψDas well.
Definition 1.1. For i ∈˜I and k ∈ I we define effective divisors E±
by
i and F±
k
E+
i=
Nσy(i)
?
σx(i)−1
?
j=σx(i)+1
2
Dσy(i),j,F+
k=
k−1
?
N−1
?
2
i=1
2
E+
i,
E−
i=
2
j=0
Dσy(i),j,F−
k=
2
i=k+1
2
E−
i.
Example 1.2. Let us consider as an example the case N0= 4, N1= 2 and
(σ(i))i∈˜I=
??7
2,0
?
,
?3
2,1
?
,
?5
2,0
?
,
?3
2,0
?
,
?1
2,1
?
,
?1
2,0
??
.
We show the corresponding diagram Γσ in Figure 1. The divisors are given as
follows:
E+
1
2:=
?
?
?
?
?
?
0
0
0
0
0
001
?
?
?
?
?
?
,F+
1:=
?
?
?
?
?
?
0
0
0
0
0
001
?
?
?
?
?
?
,
E+
3
2:=
0
0
0
0
1
000
,F+
2:=
0
0
0
0
1
001
,
E+
5
2:=
0
0
0
0
0
011
,F+
3:=
0
0
0
0
1
012
,
E+
7
2:=
0
0
0
0
0
111
,F+
4:=
0
0
0
0
1
123
,
E+
9
2:=
0
0
1
0
1
000
,F+
5:=
0
0
1
0
2
123
,
E+
11
2
:=
0
0
0
1
0
111
,F+
6:=
0
0
1
1
2
234
.
Here the (ε,x)-th matrix element represent the coefficient of the divisor Dε,x.
Lemma 1.3.
(1) OYσ(E+
i+ E−
i) ≃ OYσ,
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Figure 1: Γσ
(2) OYσ(F+
Proof. We have
k) ≃ OYσ(F−
k).
ψE+
i+E−
i=
?
y − z
−y
(σx(i) = 0),
(σx(i) = 1),
so the equation (1) follows. Now, for the equation (2) it is enough to show that
the deviser
N−1
?
gives the trivial bundle. In fact, we have ψF+
F+
N:=
2
i=1
2
E+
i
N= −x.
We denote the line bundle OYσ(F+
k) ≃ OYσ(F−
k) on Yσby Lk. We set
F±=
N−1
?
k=1
F±
k,L =
N−1
?
k=1
Lk.
Example 1.4. In the case Example 1.2,
F+=
?
0
0
1
0
5
25 10
?
.
Lemma 1.5. For i ∈˜I we have
F+
?
σy(i),σx(i) +1
2
?
− F+
?
σy(i),σx(i) −1
?
2
?
= F+
?
σy(i + 1),σx(i + 1) +1
2
− F+
?
σy(i + 1),σx(i + 1) −1
2
?
+ 1.
Proof. First, note that
E+
j
?
σy(i),σx(i) +1
2
?
− E+
j
?
σy(i),σx(i) +1
2
?
= δi,j.
So we have
F+
k
?
σy(i),σx(i) +1
2
?
− F+
k
?
σy(i),σx(i) +1
2
?
=
?
0
1
(k < i),
(k > i),
and so
F+
?
σy(i),σx(i) +1
2
?
− F+
?
σy(i),σx(i) −1
2
?
= N − i −1
2.
Thus the claim follows.
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Proposition 1.6. The line bundles L is generated by its global sections.
Proof. It is enough to prove that the support function ψF+ is upper convex
([Ful93, §3.4]). It is enough to prove that ψF+ is upper convex on Tk−1
for any k ∈ I. We denote the edge which is the intersection of Tk−1
by lk. The configurations of lk, Tk−1
two cases:
2∩Tk+1
2and Tk+1
2
2
2and Tk+1
2are classified into the following
(1) The union of Tk−1
this case, the point (σx(k +1
lk−1, the point (σx(k −1
(2) The union of Tk−1
this case, the point (σx(k +1
σx(k −1
In both cases it follows from Lemma 1.5 that ψF+ is upper convex on Tk−1
Tk+1
2and Tk+1
2is a parallelogram and lkis its diagonal. In
2)+1
2) +1
2,σy(k +1
2,σy(k −1
2)) is the intersection of lkand
2)) is the other end of lk−1.
2and Tk+1
2is a triangle and lk is its median line. In
2) +1
2) + 1, σy(k −1
2,σy(k +1
2) = σy(k +1
2)) is the middle point and
2).
2) = σx(k +1
2∩
2.
Given a divisor D the space of global sections of the line bundle OYσ(D) is
described as follows:
H0(Yσ,OYσ(D)) ≃
?
u∈S0
∆(D)
C · eu,
where
S0
∆(D) := {u ∈ M | ?u,v? ≥ ψD(v) (∀v ∈ |∆|)}.
For u ∈ M we define
ZD(u) := {v ∈ |∆| | ?u,v? ≥ ψD(v)}.
Then the cohomology of the line bundle OYσ(D) is given as follows ([Ful93,
§3.5]):
Hk(Yσ,OYσ(D)) ≃
u∈M
?
Hk(|∆|,|∆|\ZD(u);C).
We have the exact sequence of relative cohomologies
0
→
→
H0(|∆|,|∆|\ZD(u))
H1(|∆|,|∆|\ZD(u))
→
→
H0(|∆|)
H1(|∆|)
i∗
−→
−→
H0(|∆|\ZD(u))
···
Note that H0(|∆|) = C, H1(|∆|) = 0 and if |∆|\ZD(u) is not empty then i∗
does not vanish. We define
Z◦
D(u) := {v ∈ |∆| ∩ {z = 1} | ?u,v? < ψD(v)},
then |∆|\ZD(u) is homeomorphic to Z◦
Now, in our situation it follows from the convexity of ψ−F that the number
of connected components of Z◦
−F(u) is at most 2. Let us denote
D(u) × R.
S1
∆(−F) := {u ∈ M | Z◦
−F(u) has two connected components}.
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Proof. By the argument in the proof of Proposition 3.10, Lζ
the divisor which is described as a sum of divisors of the form E+
We can verify the claim using the description (5) of H1of a line bundle on
Y .
kis associated with
i+E−
j(i < j).
Proposition 3.12. The set of the torus fixed points Mζ(v)Tis isolated.
Proof. By Corollary 3.11 we have OY ∈ Pζ.
corresponding to OY. By Proposition 3.6 and Proposition 3.8, the moduli space
Mζ(v) parametrizes finite dimensional quotient µk(A)-modules V′of Pζ with
dimV′= µk(v). Note that
Let Pζ be the µk(A)-module
(Pζ)k= H0(Y,(Lζ
k)−1)
and the T-weight decomposition of H0of a line bundle on Y is multiplicity free.
Hence the claim follows.
4 Remarks
In this section, we make some observations on how Kontsevich-Soibelman’s wall-
crossing formula (9) (they also call the formula (9) by ”Factorization Property”)
would be applied in our setting.
First, we will review the work of Kontsevich-Soibelman ([KS]) very briefly.
The core of their work is the construction of the algebra homomorphism from
the ”motivic Hall algebra” to the ”quantum torus”. For an A∞-category C,
the motivic Hall algebra H(C) is, roughly speaking, the space of motives over
the moduli Ob(C) of all objects in C, with the product derived from the same
diagram as the Ringel-Hall product. The quantum torus is a deformation of a
polynomial ring described explicitely in the terms of the numerical datum of
C. The homomorphism is given by taking, so to say, weighted Euler character-
istics with respect to the motivic weight, where the motivic weight is defined
using motivic Milnor fiber of the potentials coming from the A∞-structure. The
formula (9) is the translation of the Harder-Narashimhan property under this
homomorphism.
Remark 4.1. In the original Donaldson-Thomas invariants defined using sym-
metric obstruction theory, we adapt the Behrend’s function as a weight (see
§2.3). It is expected what, after taking the ”quasi-classical limit” as q → 1, the
motivic weight would coincide with the Behrend’s one. In [KS], the proof of
the claim for some special situations and some evidences of the claim for more
general situations are provided.
Now, we will explain the statement of ”Factorization Property”, restricting
to our situation. We set Λ := Z˜ Q0and define the skew symmetric bilinear form
?−,−?: Λ × Λ → Z by
?(ei),(fi)? := e∞· f0− e0· f∞.
Let Z ∈ Hom(Λ,C) be a homomorphism such that Im(Z(Λ+)) > 0 where Λ+=
Z
≥0.
˜ Q0
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Let Dµbe a certain ring of motives including the inverting motives L−1,
[GL(n)]−1(n ≥ 1) and the formal symbol L1/2, where L is the motive of the
affine line. We have the homomorphism of rings φ: Dµ→ Q(q1/2) mapping L1/2
to q1/2. We define the quantum torus RΛ,qas the Q(q1/2)-algebra generated by
xγ(γ ∈ Λ) with the relation
xγxµ= q
1
2?γ,µ?xγ+µ.
For a strict sector V in the upper half plane, let CZ
of˜A-modules which can be described as subsequent extensions of Z-semistable
objects E such that Z(E) ∈ V . Note that CZ
Hom(Λ,C), unless the values of Z of semistable objects get close to the boundary
∂V . We define an element AZ
V,q∈ RV,q by ”weighted” counting of objects in
CZ
?
Vdenote the category
Vdoes not change when Z moves in
V. Informally speaking,
AZ
V,q:=
?
E∈Isom(CZ
V)
φ
w(E)
[Aut(E)]
?
· xdim(E)∈ RV,q,
where w(E) ∈ Dµis defined by the motivic Milnor fiber of the potential of the
A∞-algebra algebra Ext∗(E,E).
V1
V2
Figure 6: V
Assume that V is decomposed into a disjoint union V = V1⊔ V2 in the
clockwise order. Then the ”Factorization Property” in [KS] claims that
AZ
V,q= AZ
V1,q· AZ
V2,q. (9)
The key fact is, as we mentioned above, the existence of the algebra homomor-
phism from the motivic Hall algebra to RV,q. Although the category of perverce
coherent systems is not Calabi-Yau, Proposition 2.1 would assure the existence
of the algebra homomorphism in our case. We can define an element AZ
the motivic Hall algebra and the equation AZ
from the Harder-Narashimhan property. Now the element AZ
AZ
V,motunder the algebra homomorphism.
Now we end up with reviewing and begin to explain how to apply (9) in our
setting. Since we are interested in˜A-modules V with V∞≃ C, we will work on
the quotient algebra
R′
Consider the wall in Hom(Λ,C) such that e ∈ Λ+with e∞ = 0 and f ∈ Λ+
with f∞ = 1 are send on a same half line in the upper half plane. Assume
e ∈ Λ is primitive (i.e. {ei}i∈Q0are coprime to each other) and f − e / ∈ Λ+.
V,motin
V,mot= AZ
V1,mot· AZ
V,qis the image of
V2,motfollows
Λ,q:= RΛ,q/(xe){e|e∞≥2}.
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Λ0∩ Λ+
l0
l1
l2
l3···
e∞= 1
l∞
e
f
Figure 7: Λ0
Let Λ0∈ Λ be the sublattice generated by e and f, lkbe the half line passing
through k · e + f and l∞be the half line passing through e. Take Z+and Z−
from the opposite side of the wall so that l1,l2,...,l∞are mapped on the upper
half plane in the clockwise (resp. anticlockwise) way by Z+(resp. Z−). The
”Factorization Property” claims
AZ+
l1
· AZ+
l2
· ··· · AZ+
l∞= AZ−
l∞· ··· · AZ−
l2
· AZ−
l1
in R′
Λ,q. We denote
→
?
k
AZ+
lk
:= AZ+
l1· AZ+
l2
· ··· ,
←
?
k
AZ−
lk
:= ··· · AZ−
l2
· AZ−
l1.
Note that AZ+
described as following:
l∞= AZ−
l∞and we denote this by AZ
l∞. Then the above equation is
→
?
k
AZ+
lk
= AZ
l∞·
?→
?
k
AZ−
lk
?
· AZ
l∞
−1.
An element A of R′
Λ,qcan be uniquely described in the following form:
A =
?
e; e∞=0
(ae(q) · xe) + x∞·
?
e;e∞=0
(be(q) · xe).
We denote?
e; e∞=0(be(q) · xe) by Ax∞. Then
(q − 1) ·
?→
?
k
AZ+
lk
?x∞?????
q=1
makes sense and would coincide with the generating function of virtual counting
of the moduli spaces we study in §2. Note that we have
?k · e + f,e? = e0.
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and so
xm·e· xk·e+f= qm·e0xk·e+f· xm·e.
Λ,qwith the polynomial in xe, then we have
Identify AZ+
l∞∈ R′
→
?
k
AZ+
lk
= AZ
l∞(xe) ·
?←
?
?
k
AZ−
lk
?
· AZ
l∞(xe)−1
=
?←
?
k
AZ−
lk
· AZ
l∞(qe0xe) · AZ
l∞(xe)−1
in R′
crossing formulas.
Here again, observations in [KS] will help us. We put t := xe. Assume that
we have the unique simple object E on l∞ and dimE = e. Let BE be the
algebra generated by Ext1(E,E) with relations defined from the potential WE.
Then we have
AZ
Λ,q. Now, if we can compute AZ
l∞(qe0xe) · AZ
l∞(xe)−1??
q=1, we get wall-
l∞(qt) = AZ
l∞(t) · f(t) (10)
where f(t) is obtained by counting pairs of cyclic BE-modules and their cyclic
vectors. Applying this formula repeatedly we have
AZ
l∞(qe0t) · AZ
l∞(t)−1??
q=1= (f(t)|q=1)e0.
Example 4.2.
we have f(t)|q=1 = 1 + t. This corresponds to the formula in Theorem
2.14.
(1) Assume ext1(E,E) = 0. The algebra BEis trivial. Hence
(2) Assume ext1(E,E) = 1 and BE≃ C[z]. In this case we have f(t)|q=1=
(1 − t)−1. This corresponds to the formula in Theorem 2.18.
(3) Let us consider the wall corresponding to the imaginary root. The set of
simple objects on this wall is {Oy| y ∈ Y }. By the same argument as they
show the above equation (10) in [KS], we would have
AZ
l∞(qt) = AZ
l∞(t) · f(t)
where f(t) is obtained by counting 0-dimensional closed subscheme of Y .
By the results of [MNOP06] and [BF] we have
f(t)|q=1= M(−t)e(Y )
where
M(t) :=
∞
?
n=1
(1 − tn)−n
is the MacMahon function. This provides DT-PT correspondence in our
situation.
39
Page 40
References
[Beh]K. Behrend, Donaldson-Thomas invariants via microlocal geometry,
math.AG/0507523.
[BF]K. Behrend and B. Fantechi, Symmetric obstruction theories and
Hilbert schemes of points on threefolds, math.AG/0512556.
[Boc08]R. Bocklandt, Graded Calabi Yau algebras of dimension 3, J. Pure
Appl. Algebra 212 (2008), no. 1, 14–32.
[Bri02]T. Bridgeland, Flops and derived categories, Invent. Math. 147
(2002), no. 3, 613–632.
[CB02]W. Crawley-Boevey, Decomposition of Marsden-Weinstein reduc-
tions for representations of quivers, Comp. Math. 130 (2002), no. 2,
225–239.
[Ful93]W. Fulton, Introduction to toric varieties, Annals of Mathematics
studies, Princeton University Press, 1993.
[GV]R. Gopakumar and C. Vafa, M-theory and topological strings II,
hep-th/9812127.
[HST01]S. Hosono, M. Saito, and A Takahashi, Relative Lefschetz actions
and BPS state counting, Internat. Math. Res. Notices 15 (2001),
783–816.
[IKP] A. Iqbal and A-K Kashani-Poor, The vertex on a strip, hep-
th/0410174.
[IU]A. Ishii and K. Ueda, On moduli spaces of quiver representations
associated with brane tilings, arXiv:0710.1898.
[Joy]D. Joyce, Configurations in abelian categories IV. Invariants and
changing stability conditions, math/0410268.
[Kin94]A.D. King, Moduli of renresentations of finite dimensional algebras,
J. Algebra. 45 (1994), no. 4, 515–530.
[KS]M. Kontsevich and Y. Soibelman,
tivic Donaldson-Thomas invariants and cluster transformations,
arXiv:0811.2435.
Stability structures, mo-
[Kuz]A.
AG/0111092.
G. Kuznetsov,Quiver varietiesandHilbert schemes,
[LP93]J. Le Potier, Syst` emes coh´ erents et structures de niveau, Ast´ erisque
(1993), no. 214, 143 pp.
[MNOP06] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande,
Gromov-Witten theory and Donaldson-Thomas theory, I, Comp.
Math. 142 (2006), 1263–1285.
[MR]S. Mozgovoy and M. Reineke, On the noncommutative donaldson-
thomas invariants arising from brane tilings, arXiv:0809.0117.
40
Page 41
[Nak94]H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-
Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416.
[Nak98]
, Quiver varieties and Kac-Moody algebras, Duke Math. J.
91 (1998), no. 3, 515–560.
[Nak99]
, Lectures on Hilbert schemes of points on surfaces, Univer-
sity Lecture Series, American Mathematical Society, Providence, RI,
1999.
[Nak01]
, Quiver varieties and finite-dimensional representations of
quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–
238.
[NN] K. Nagao and H. Nakajima, Counting invarinats of perverse coher-
ent systems on 3-folds and their wall-crossings, arXiv:0809.2992.
[NYa]H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-
up. I. a quiver description, arXiv:0802.3120.
[NYb]
, Perverse coherent sheaves on blow-up. II. wall-crossing and
Betti numbers formula, arXiv:0806.0463.
[PTa] R. Pandharipande and R.P. Thomas, Curve counting via stable pairs
in the derived category, arXiv:0707.2348.
[PTb]
, Stable pairs and BPS invariants, arXiv:0711.3899.
[Rud97] A. Rudakov, Stability for an Abelian category, J. Algebra. 197
(1997), no. 1, 231–245.
[Sze]B. Szendroi, Non-commutative Donaldson-Thomas theory and the
conifold, arXiv:0705.3419v3.
[Tho00]R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-
folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000),
no. 2, 367–438.
[Tod] Y. Toda, Birational Calabi-Yau 3-folds and BPS state counting,
arXiv:0707.1643.
[VdB]M.
math/0211064.
Vanden Bergh,Non-commutative crepantresolutions,
[VdB04]
, Three-dimensional flops and noncommutative rings, Duke
Math. J. 122 (2004), no. 3, 423–455.
[Youa]B. Young, Computing a pyramid partition generating function with
dimer shuffling, arXiv:0709.3079.
[Youb]J Young, B. with an appendix by Bryan, Generating functions for
colored 3d Young diagrams and the Donaldson-Thomas invariants
of orbifolds, arXiv:0802.3948.
Kentaro Nagao
Research Institute for Mathematical Sciences, Kyoto University, Kyoto
606-8502, Japan
kentaron@kurims.kyoto-u.ac.jp
41