Derived categories of small toric Calabi-Yau 3-folds and counting invariants

Abstract

We first construct a derived equivalence between a small crepant resolution
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 systems. As an application we provide certain equations on
Donaldson-Thomas, Pandeharipande-Thomas and Szendroi's invariants. 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.

Full text

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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.
1 Derived equivalences
1.1tilting 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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