Gromov-Witten theory and Donaldson-Thomas theory, II

0.24 MB·English

by  D. Maulik

Gromov-Witten theory and
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0 Donaldson-Thomas theory, II
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J D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande

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1 July 2004
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1 Introduction
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1.1 Overview
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The Gromov-Witten theory of a 3-fold X is defined via integrals over the
v
2 moduli space of stable maps. The Donaldson-Thomas theory of X is defined
9 via integrals over the moduli space of ideal sheaves. In [14], a GW/DT
0
correspondence equating the two theories was proposed, and the Calabi-Yau
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0 case was presented. We discuss here the GW/DT correspondence for general
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3-folds.
0
/ Let X be a nonsingular, projective 3-fold. Insertions in the Gromov-
h
t Witten theory of X are determined by primary and descendent fields. Inser-
a
m tions in the Donaldson-Thomas theory of X are naturally obtained from the
: Chern classes of universal sheaves. We conjecture a GW/DT correspondence
v
for 3-folds relating these two sets of insertions.
i
X
Let S ⊂ X be a nonsingular surface. The Gromov-Witten theory of
r
a X relative to S has been defined in [4, 9, 10, 12]. The relative constraints
are determined by partitions weighted by cohomology classes of S. A rela-
tive Donaldson-Thomas theory has been defined by J. Li [13]. The relative
constraints are determined by cohomology classes of the Hilbert scheme of
points of S. We propose a GW/DT correspondence in the relative case re-
lating the Gromov-Witten constraints to the Donaldson-Thomas constraints
via Nakajima’s basis of the cohomology of the Hilbert scheme of points.
InthelastSectionofthepaper,independentoftheconjecturalframework,
we study the Donaldson-Thomas theory in degree 0 using localization and
1
relative geometry. We derive a formula for the equivariant vertex measure in
the degree 0 case and prove Conjecture 1′ of [14] in the toric case. A degree
0 relative formula is also proven.
1.2 Acknowledgments
We thank J. Li for explaining his definition of relative Donaldson-Thomas
theory to us. An outline of his ideas is presented in Section 3.2.1. We thank
J. Bryan, T. Graber, A. Iqbal, M. Kontsevich, Y. Soibelman, R. Thomas,
and C. Vafa for related discussions.
D. M. was partially supported by a Princeton Centennial graduate fel-
lowship. N. N. was partially supported by the grants RFFI 03-02-17554 and
NSh-1999.2003.2. He is grateful to the Princeton Mathematics department
for hospitality. A. O. was partially supported by DMS-0096246 and fellow-
ships from the Sloan and Packard foundations. R. P. was partially supported
by DMS-0071473 and fellowships from the Sloan and Packard foundations.
2 The GW/DT correspondence for 3-folds
2.1 GW theory
Gromov-Witten theory is defined via integration over the moduli space of
stable maps. Let X be a nonsingular, projective 3-fold. Let M (X,β)
g,r
denote the moduli space of r-pointed stable maps from connected, genus g
curves to X representing the class β ∈ H (X,Z). Let
2
ev : M (X,β) → X,
i g,r
L → M (X,β)
i g,r
denote the evaluation maps and cotangent lines bundles associated to the
marked points. Let γ ,...,γ be a basis of H∗(X,Q), and let
1 m
ψ = c (L ) ∈ M (X,β).
i 1 i g,n
The descendent fields, denoted by τ (γ ), correspond to the classes ψkev∗(γ )
k i i i j
on the moduli space of maps. Let
r
hτ (γ )···τ (γ )i = ψkiev∗(γ )
k1 l1 kr lr g,β i i li
Z[Mg,r(X,β)]vir i=1
Y
2
denote the descendent Gromov-Witten invariants. Foundational aspects of
the theory are treated, for example, in [1, 2, 11].
Let C be a possibly disconnected curve with at worst nodal singularities.
′
The genus of C is defined by 1 − χ(O ). Let M (X,β) denote the mod-
C g,r
uli space of maps with possibly disconnected domain curves C of genus g
with no collapsed connected components. The latter condition requires each
connected component of C to represent a nontrivial class in H (X,Z). In
2
particular, C must represent a nonzero class β.
The descendent invariants are defined in the disconnected case by
r
hτ (γ )···τ (γ )i′ = ψkiev∗(γ ).
k1 l1 kr lr g,β ′ i i li
Z[Mg,r(X,β)]vir i=1
Y
Define the following generating function,
r r
Z′ X;u | τ (γ ) = h τ (γ )i′ u2g−2. (1)
GW ki li ki li g,β
β
(cid:16) Yi=1 (cid:17) Xg∈Z Yi=1
Since the domain components must map nontrivially, an elementary argu-
ment shows the genus g in the sum (1) is bounded from below. The descen-
dent insertions in (1) should match the (genus independent) virtual dimen-
sion,
dim [M′ (X,β)]vir = c (T )+r.
g,r 1 X
Zβ
Following the terminology of [14], we view (1) as areduced partition function.
2.2 DT theory
Donaldson-Thomas theory is defined via integration over the moduli space
of ideal sheaves. Let X be a nonsingular, projective 3-fold. An ideal sheaf
is a torsion-free sheaf of rank 1 with trivial determinant. Each ideal sheaf I
injects into its double dual,
0 → I → I∨∨.
As I∨∨ is reflexive of rank 1 with trivial determinant,
I∨∨ =∼ O ,
X
3
see [17]. Each ideal sheaf I determines a subscheme Y ⊂ X,
0 → I → O → O → 0.
X Y
The maximal dimensional components of Y (weighted by their intrinsic mul-
tiplicities) determine an element,
[Y] ∈ H (X,Z).
∗
Let I (X,β) denote the moduli space of ideal sheaves I satisfying
n
χ(O ) = n,
Y
and
[Y] = β ∈ H (X,Z).
2
Here, χ denotes the holomorphic Euler characteristic.
TheDonaldson-Thomasinvariantisdefinedviaintegrationagainstvirtual
class,
[I (X,β)]vir.
n
Foundational aspects of the theory are treated in [15, 20].
Lemma 1. The virtual dimension of I (X,β) equals c (T ).
n β 1 X
Proof. The virtual dimension, obtained from the obstrRuction theory, is
χ(O ,O )−χ(I,I),
X X
where
3
χ(A,B) = (−1)idimExti(A,B).
i=0
X
Since X is a nonsingular 3-fold, there exists a finite resolution of I by locally
free sheaves,
0 → F → F → F → F → I → 0.
3 2 1 0
Let x denote the Chern roots of F . Since the determinant of I is trivial,
ij i
3
(−1)ix = 0.
ij
i=0 j
XX
4
Since the fundamental class of Y is β,
−ch (I) = ch (O ) = β.
2 2 Y
We will calculate the virtual dimension in terms of the Chern roots via
GRR. The first term is,
χ(O ,O ) = Td(X). (2)
X X
ZX
Next,
3 3
−χ(I,I) = − (−1)ie−xij · (−1)ˆiexˆiˆj ·Td(X).
ZX (cid:0)Xi=0 Xj (cid:1) (cid:0)Xˆi=0 Xˆj (cid:1)
SincetheChernrootexpression intheintegrandiseven,onlythecomponents
in degrees 0 and 2 need be considered. The degree 0 component is equal to 1,
the square of the rank of I. The integral of the degree 0 component against
Td(X) cancels the first term (2). The degree 2 component is
3 x2 x2 3
(−1)i+ˆi ij −x x + ˆiˆj = 2ch (I)− (−i)i+ˆix x .
2 ij ˆiˆj 2 2 ij ˆiˆj
!
iX,ˆi=0Xj,ˆj iX,ˆi=0Xj,ˆj
The second term on the right equals the square of the determinant of I and
hence vanishes. We conclude the virtual dimension equals
− 2ch (I)·Td(X) = c (X)
2 1
ZX Zβ
since the degree 1 term of Td(X) is c (X)/2.
1
ThemodulispaceI (X,β)iscanonicallyisomorphictotheHilbertscheme
n
[15]. As the Hilbert scheme is a fine moduli space, universal structures are
well-defined. Let π and π denote the projections to the respective factors
1 2
of I (X,β)×X. Consider the universal ideal sheaf I,
n
I → I (X,β)×X.
n
Since I is π -flat and X is nonsingular, a finite resolution of I by locally free
1
sheaves onI (X,β)×X exists. Hence, theChernclasses ofIarewell-defined.
n
5
For γ ∈ Hl(X,Z), let ch (γ) denote the following operation on the
k+2
homology of I (X,β):
n
ch (γ) : H (I (X,β),Q) → H (I (X,β),Q),
k+2 ∗ n ∗−2k+2−l n
ch (γ) ξ = π ch (I)·π∗(γ)∩π∗(ξ) .
k+2 1∗ k+2 2 1
We define descenden(cid:0)t (cid:1)fields in(cid:0) Donaldson-Thomas th(cid:1)eory, denoted by
τ˜ (γ), to correspond to the operations (−1)k+1ch (γ). The descendent
k k+2
invariants are defined by
r
hτ˜ (γ )···τ˜ (γ )i = (−1)ki+1ch (γ ),
k1 l1 kr lr n,β ki+2 li
Z[In(X,β)]vir i=1
Y
where the latter integral is the push-forward to a point of the class
(−1)k1+1ch (γ ) ◦ ··· ◦ (−1)kr+1ch (γ ) [I (X,β)]vir .
k1+2 l1 kr+2 lr n
(cid:16) (cid:17)
A similar slant product construction can be found in the Donaldson the-
ory of 4-manifolds. Since the Chern character contains denominators, the
descendent invariants in Donaldson-Thomas theory are rational numbers.
Define the Donaldson-Thomas partition function with descendent inser-
tions by
r r
Z X;q | τ˜ (γ ) = h τ˜ (γ )i qn. (3)
DT ki li ki li n,β
β
(cid:16) Yi=1 (cid:17) Xn∈Z Yi=1
An elementary argument shows the charge n in the sum (3) is bounded from
below. As before, the descendent insertions in (3) should match the virtual
dimension.
The reduced partition function is obtained by formally removing the de-
gree 0 contributions,
r ZDT X;q | ri=1τ˜ki(γli)
Z′ X;q | τ˜ (γ ) = β.
DT ki li β (cid:16) ZDT(QX;q)0 (cid:17)
(cid:16) Yi=1 (cid:17)
The degree 0 partition function is determined by a conjecture of [14].
Following the Calabi-Yau case, we conjecture the series Z′ to be a rational
DT
function of q.
6
Conjecture 1. The degree 0 Donaldson-Thomas partition function for a
3-fold X is determined by:
ZDT(X;q)0 = M(−q) Xc3(TX⊗KX),
R
where
1
M(q) =
(1−qn)n
n≥1
Y
is the McMahon function.
Conjecture 2. The reduced series Z′ X;q | r τ˜ (γ ) is a rational
DT i=1 ki li β
function of q.
(cid:0) Q (cid:1)
2.3 Primary fields
The GW/DT correspondence is easiest to state for the primary fields τ (γ)
0
and τ˜ (γ).
0
Conjecture 3. After the change of variables eiu = −q,
r r
(−iu)d Z′ X;u | τ (γ ) = (−q)−d/2 Z′ X;q | τ˜ (γ ) ,
GW 0 li DT 0 li
! !
Yi=1 β Yi=1 β
where d = c (T ).
β 1 X
ConjectRure3isconsistent withthecalculationofdegeneratecontributions
in [18]. Let C be a nonsingular, genus g curve in X which rigidly intersects
cycles dual to the classes γ , ...γ . The local Gromov-Witten series is
l1 lr
determined in [18],
r 2g−2+d
sin(u/2)
Z′ X;u | τ (γ ) = u2g−2,
GW 0 li u/2
!
Yi=1 [C] (cid:18) (cid:19)
The local Donaldson-Thomas series is then predicted by Conjecture 3,
r eiu/2 −e−iu/2 2g−2+d
Z′ X;q | τ˜ (γ ) = (−iu)d(−q)d/2 u2g−2
DT 0 li iu
!
Yi=1 [C] (cid:18) (cid:19)
= q1−g(1+q)2g−2+d
7
The normalizations and signs in Conjecture 3 are fixed by the requirement
that the reduced partition function Z′ has initial term q1−g corresponding
DT
to the ideal of C.
If the cohomology classes γ are integral, the Donaldson-Thomas invari-
i
ants for primary fields are integer valued. The integrality constraints for
Gromov-Witten theory obtained via the GW/DT correspondence for pri-
mary fields were conjectured previously in [18, 19].
2.4 Descendent fields
For fixed curve class β, consider the full set of (normalized) reduced partition
functions,
Z′ = (−iu)d− ki Z′ X;u | τ (γ ) ,
GW,β GW ki li
(cid:26) P β(cid:27)
(cid:16) Y (cid:17)
where d = c (T ) as before. Here, Z′ consists of the finite set of
β 1 X GW,β
descendent series with insertions of the correct dimension. The set Z′ is
R GW,β
partially ordered by k , the descendent partial ordering. Similarly, let
i
P
Z′ = (−q)−d/2 Z′ X;q | τ˜ (γ ) .
DT,β DT ki li
β
(cid:26) (cid:27)
(cid:16) Y (cid:17)
Conjecture 4. After the change of variables eiu = −q,
(i) the sets of functions Z′ and Z′ have the same linear spans,
GW,β DT,β
(ii) there exists a canonical matrix expressing the functions Z′ as linear
GW,β
combinations of the functions Z′ :
DT,β
(a) the matrix coefficients depend only upon the classical cohomology
of X and universal series,
(b) the matrix is unipotent and upper-triangular with respect to the
descendent partial ordering.
By Conjecture 4, each element of Z′ is a canonical linear combination,
GW,β
(−iu)d− ki Z′ τ (γ ) = (−q)−d/2 Z′ τ˜ (γ ) +..., (4)
GW ki li DT ki li
P β β
(cid:16)Y (cid:17) (cid:16)Y (cid:17)
8
where the omitted terms are strictly lower in the partial ordering.
We do not yet have a complete formula for the canonical matrix of Con-
jecture 4. However, for the descendents of the point class [P] ∈ H6(X,Z),
we can formulate a precise conjecture.
Conjecture 4′. After the change of variables eiu = −q,
(−iu)d− kj Z′ τ (γ ) τ (P) =
GW 0 li kj
P β
(cid:16)Y Y (cid:17)
(−q)−d/2 Z′ τ˜ (γ ) τ˜ (P) ,
DT 0 li kj
β
(cid:16)Y Y (cid:17)
if codim(γ ) > 0 for each i.
li
2.5 Reactions
We believe the upper-triangular matrix of Conjecture 4 is determined by two
types of reactions:
τ (γ ) → A (γ ) τ (c (T )∪γ )
a l a l a−1 1 X l
τa(γl)τa′(γl′) → Aa,a′(γl,γl′) τa+a′−1(γl ∪γl′)
Thelinearcombination(4)shouldbegeneratedbyapplyingthetwo reactions
to the Gromov-Witten insertions
τ (γ )
ki li
Y
toexhaustion andthen interpreting theoutput in Donaldson-Thomastheory.
For example,
(−iu)d−k Z′ (τ (γ )) =
GW k l β
k j
(−q)−d/2 A (c (T )i−1 ∪γ ) Z′ τ˜ (c (T )j ∪γ ) .
k−i+1 1 X l DT k−j 1 X l
!
j=0 i=1
X Y (cid:0) (cid:1)
The resulting matrix will be upper-triangular with respect to the reaction
partial ordering, a refinement of the descendent partial ordering. We further
speculate that the reaction amplitudes,
Aa(γl), Aa,a′(γl,γl′),
9
aregivenbyuniversalformulasdependingonlyupontheclassicalcohomology
of X (including possibly the Hodge decomposition). Conjectures 3, 4, and 4′
are all consequences of the reaction view of the GW/DT correspondence for
descendent fields.
2.6 An example
LetX beP3 andletβ betheclass[L]ofaline. AGromov-Wittencalculation
using localization and known Hodge integral evaluations yields the following
result,
sin(u/2)
Z′ X;u | τ (L)τ (P) = cos(u/2)u−2,
GW 0 1 [L] u/2
(cid:18) (cid:19)
(cid:16) (cid:17)
see [7, 6]. By Conjecture 4′,
sin(u/2)
Z′ X;q | τ˜ (L)τ˜ (P) = (−iu)3(−q)2 cos(u/2)u−2
DT 0 1 [L] u/2
(cid:16) (cid:17) (cid:16) (cid:17)
eiu/2 −e−iu/2eiu/2 +e−iu/2
= (−iu)3(−q)2 u−2
iu 2
1
= q(1−q2)
2
The resulting Donaldson-Thomas series can be checked order by order in q
via localization.
3 The GW/DT correspondence for relative
theories
3.1 GW theory
Let X be a nonsingular, projective 3-fold and let S ⊂ X be a nonsingular
divisor. The Gromov-Witten theory of X relative to S has been defined in
[4, 9, 10, 12]. Let β ∈ H (X,Z) be a curve class satisfying
2
[S] ≥ 0.
Zβ
10