Spin structure on moduli space of sheaves on Calabi-Yau threefold

0.28 MB·English

by  Zheng Hua

Spin structure on moduli space of sheaves on
Calabi-Yau threefold
3
1
0
2 Zheng Hua ∗†

n
January 10, 2013
a
J

9

] Abstract
G
A Kontsevich and Soibelman defined a notion of orientation data on
Calabi-Yaucategory. Itcanbeviewedasaconsistentchoiceofspinstruc-
.
h tureonmodulispaceofobjectsinthegivenCYcategory. Theorientation
t data plays an important role in Donaldson-Thomas theory. Let X be a
a
m compactCY3-foldsatisfyingappropriatetopologicalcondition. Weprove
the existence and uniqueness of orientation data on the derived category
[ of coherent sheavesDb(X).

2
v
1 Introduction
0
9
7 The goal of this paper is to construct C∞ orientation data (See Definition 7.2)
3 on moduli spaces of coherent sheaves on Calabi-Yau threefolds. Orientation
.
2 data is introduced by Kontsevich and Soibelman in [12]. Roughly speaking,
1 an orientation data on a Calabi-Yau category (defined in section 3.3 of [12])
2 C
is a choice of square root of the determinant line bundle on the moduli space
1
of objects, satisfying an additional compatibility condition. Depending on the
:
v structure we put on the determinant line bundle, one can consider different
i
X types of square roots. In the original definition of Kontsevich and Soibelman,
they consider the determine bundle as a (ind-)constructible super line bundle
r
a overthemodulispaceofallobjectsin . Wewillnotstudyinsuchagenerality.
C
Let be Db(coh(X)), the derived category of coherent sheaves on a smooth
C
projective CY 3-fold X. Moreover, we will restrict to moduli stack of coherent
sheaves, denoted by ( for short), instead of moduli of arbitrary com-
X
M M
plexes. One can stratify such that over each strata there is a graded vector
bundlewithfiberExt• (EM,E)atpointE . Byabusingnotations,wedenote
this(ind-)constructibXlegradedvectorbun∈dlMebyExt• (E,E). Inparticular,this
X
means rank of Exti (E,E) is constant for all i over each strata. We define the
X
(ind-)constructible determinant line bundle L to be sdet(Ext• (E,E)) where
X
∗DepartmentofMathematics, KansasStateUniversity
†Institute of Mathematical Science, Chinese University of Hong Kong
Email:[email protected]
1
sdet is the super determinant. Kontsevich-Soibelman’s orientation data is a
choice of (ind-)constructible square root of L that is compatible under Hall
multiplication. The orientation data is a critical ingredient in the definition of
motivic Donaldson-Thomas invariant (See section 6 [12]).
The (ind-)constructible structure is the weakest structure that one can put
ondeterminantlinebundle. However,itissufficientforthepurposeof[12]since
motivicDTinvariantsaredefinedbyfirststratifythemodulistackandthenadd
the different strata with appropriate motivic weights. By the work of Quillen
and Knudsen-Mumford, the determinant line bundle can be equipped with a
holomorphic or algebraic structure, without stratifying the moduli space. We
will review these constructions in Section 2. Given the holomorphic determi-
nant line bundle L over the underlying analytic stack associated to , a C∞
orientation data is a choice of square root of L that is compatible uMnder Hall
multiplication(SeeDefinition7.2). Fromthepointofview ofderivedgeometry,
the determinant line bundle L plays the role of canonical bundle of moduli
space. We call a square root of L a spin structure on .
The existence and classification of C∞ orientationMdata on moduli space is
a purely topological question, essentially determined by the homotopy type of
X. Inthispaper,weconsideraspecialclassofCY3-foldscalledadmissible CY
3-folds (Definition 6.8). The main theorem of the paper is:
Theorem1.1. (Theorem 7.6)IfX isanadmissibleCY3-fold, thenthereexists
a C∞ orientation data on Db(X).
Admissible condition is a condition on torsion part of homology of X. Any
simply connected torsion free CY threefold is admissible (Theorem 6.10).
The proof of Theorem 7.6 is a combination of gauge theory and surgery
theory. Thesametechniquehasbeenusedwidelyinthestudy offourmanifold.
A very good reference is chapter 5 of [6]. In the first step, we reduce the
question about moduli space of coherent sheaves to moduli of vector bundles.
This is essentially consequence of a theorem of Joyce and Song (Theorem 7.3).
Using gauge theory we realize the moduli space of holomorphic vector bundles
as analytic subspace of a complex Banach manifold modulo automorphisms.
A simple but important observation is that the determinant line bundle L
extends to the ambient Banach manifold. Instead of constructing square root
of L on the analytic subspace,we do it onthe ambientBanachmanifold. This
is equivalent with proving c (L) is divisible by two. Over rational number,
1
it follows from Grothendieck-Riemann-Roch theorem and Atiyah-Singer index
theorem(Theorem3.1,Theorem5.1). Thetorsionpartofc (L)isnotcaptured
1
by GRR or AS. We get around it by proving the even torsion cannot occur if
X is admissible. The odd torsion never matters anway. To be more specific,
we consider those CY 3-folds that can be uniformized to connected sum of
S3 S3 by a sequence of conifoldtransitions and study how the torsionpartof
×
the secondcohomologygroupofthe Banachmanifoldmentionedabovechanges
under the conifold transition.
Moduli space of sheaves on CY 3-fold is locally an intersection of two holo-
morphicLagrangiansubvarietiesinsideholomorphicsymplecticmanifold. Donaldson-
2
Thomas invariant is the (weighted) sum of Lagrangian intersection numbers.
There are two generalizations of DT invariants. The first one is to replace the
DTinvariants,whicharenumbers,byelementsinappropriateGrothendieckring
ofstacks. Suchgeneralization,due to Joyce-SongandKontsevich-Soibelman,is
calledthe motivic DT invariants([11][12]). Thesecondwayisto replaceitby a
perversesheaf on . Such perversesheafis knownto existlocally by the work
M
of Joyce and Song [11]. However, the gluing problem of these locally defined
perversesheavesis nontrivial. We refer to [3] and [13] for some recent progress.
In both of these two generalized DT theories, the orientation data, i.e. spin
structure on plays an essential role.
M
This paper is organized as follows. In section 2, we recall two definitions
of determinant line bundle. In section 3 and 5, we prove the rational c (L) is
1
divisible by two. In section4, we recallsome results in topology of loopspaces.
In section 6, we define admissible CY 3-fold study the torsion in cohomology
of space of principal bundles. The main theorem 7.6 is proved in Section 7. In
the lastsection,weexplainalinkbetweenorientationdataandvolumeformon
Lagrangiandistributionwhichis usedto categorifyDT invariants. This section
is independent from other sections. The readers might consider reading it first
to get some motivations.
Acknowledgement Theexistenceoforientationdataisprovedinthecaseof
representations of quiver with potential by Davison [5]. Theorem 3.1 is known
to Kontsevich. Thanks to Dominic Joyce for pointing out that. I would like to
thank Conan Leung, Dominic Joyce for some very helpful discussion. Thanks
to Prof. Simon Donaldson for his comments about η-invariants. Thanks to
Young-HoonKiem for pointing out a mistake in the earlier version. Partof the
work was done during my visit to Oxford. I am grateful to Balazs Szendroi for
his hospitality.
2 Determinant line bundle
We recall two definitions of determinant line bundle, one in algebraic geometry
and one in differential geometry. Both definitions will be used later.
Let ( for short) be the moduli space of sheaves on X. This is an
X
M M
Artin stack locally of finite type. One can write as disjoint union indexed
M
by classes in topological K-theory:
= .
β
M M
β∈KG0(X)
2.1 Algebraic definition
Let be the universal sheaf over X. Denote the projection X
E M× M× →M
by π.
3
Definition 2.1. The (algebraic)determinant line bundle L over is defined
M
to be
sdet(π RHom( , )[1])
∗
E E
where RHom is the sheaf derived endomorphism.
Because isflatover andX issmooth,RHom( , )isaperfectcomplex.
E M E E
The determinant is well defined.
Let E be a sheaf over X. Its equivalence class [E] represents a point in the
moduli stack . The tangent complex of at [E] is defined to be the graded
M M
vector space
T = Exti (E,E).
[E]M X
Mi
IthasanobviousgradedLiealgebrastructurebyanti-commutingtheassociative
product. The degree zero piece is the subalgebra of endomorphisms whose
corresponding group is Aut(E).
The fiber of the determinant line bundle L at [E] is
top top
( Exteven(E,E))−1 Extodd(E,E).
⊗
^ ^
It inherits an action of Aut(E).
AsheafE iscalledsimple ifExt0(E,E)=C. Wedenotethemodulispaceof
∼
simple sheavesby si. It is a C∗ gerb overa scheme (Deligne-Mumford stack)
M
locally of finite type. By restricting to sheaves with fixed determinant we can
removethis C∗ gerbconsistently and obtain a moduli scheme (DM stack). The
determinant line bundle descends.
2.2 Analytic definition
WegiveaseconddefinitionofL formodulispaceofholomorphicvectorbundles.
Our main reference is [7] and [15]
LetE beacomplexvectorbundleoverX. Weidentifyitwiththeframebun-
dleofaprincipalGc bundleforGc =GL(n,C)wherenistherankofE. Denote
A for the infinite dimensional linear manifold A0,1(ad E) of (0,1) forms with
value inadjointbundle. An infinite dimensionalgaugegroupGc :=C∞(X,Gc)
acts on A. The orbit space A/Gc parameterizes equivalence classes of (0,1)-
connections. Typically, we complete A to a complex Banach space under ap-
propriate Sobolev norm. However, the results in this paper are insensitive to
particular choice of Sobolev norm. So we will ignore the completion.
A (0,1) connection :A0,0(E) A0,1(E) is called integrable if its curva-
ture F = 2 vanishes∇. Strictly spe→aking, A is identified with the underlying
∇
affine space∇of A0,1(ad E). Given two (0,1)-connections A and A′, their differ-
ence A A′ lies in A0,1(ad E).
−
If we fix an integrable reference connection , a connection := +A
A
for A A0,1(ad E) is integrable if and only if∇it satisfies the M∇aurer-C∇artan
∈
equation
d A+A A=0
∇
∧
4
where d := [ , ]. The space of integrable connections, denoted by A(1,1),
∇
is an analytic s∇ub−space of A. Because integrable (0,1) connections are in one
to one correspondence with holomorphic structures on E, the Banach analytic
stackA(1,1)/Gcparameterizesequivalenceclassesofholomorphicvectorbundles
with underlying complex vector bundle being E.
If we pick a hermitian metric on E, then the gauge group Gc is reduced
to the unitary group G = U(n). Similar to the fixing determinant trick in
algebraicgeometry,wecanconsiderthegaugegrouptobespecialunitarygroup.
By abusing the notations, we use E to denote the underlying principal SU(n)
bundle.
Given a connection A, there is a first order elliptic operator D acting on
A
the Dolbeault complex L:=A0,•(adE), defined as
D = + ∗ :Leven Lodd
A ∇A ∇A →
where ∗ istheadjointwithrespecttotheL2 metricinducedbythehermitian
∇A
metric on E. The adjoint D∗ maps Lodd to Leven.
A
When is integrable, there are isomorphisms
A
∇
Ker DA ∼=Exteven(E∇A,E∇A),Ker DA∗ ∼=Extodd(E∇A,E∇A)
where E is the corresponding holomorphic vector bundle. When is non-
∇A ∇A
integrable, D is invertible.
A
Definition 2.2. For the family ofelliptic operatorsD with A A, we define
A
the fiber of its determinant line bundle L at A to be ∈
top top
( Ker D )−1 Ker D∗.
A ⊗ A
^ ^
One can turn L into a holomorphic line bundle over A. Define the ∂¯-
Laplacian ∆ to be the second order elliptic operator D D∗. For A A and
A A A ∈
a positiverealnumber l thatis notinthe spectrumof∆ ,there exists anopen
A
neighborhood U of A such that the direct sum of eigenspaces of ∆
A A
H+ = H+
<l λ
Mλ<l
and
H− = H−
<l λ
Mλ<l
form holomorphic vector bundles over U . The determinant line bundle L is
A
defined locally to be
top top
( H+)−1 H−.
<l ⊗ <l
^ ^
Because D is an isomorphism from H+ to H− for λ positive, the above defi-
A λ λ
nition is independent with choice of l.
5
The elliptic operator D restricts to a linear transform from H+ to H−.
A <l <l
Its determinant det(D ) defines a section of L over U that vanishes exactly
A A
when Ker(DA) is nonzero. Onthe overlapof two charts UA andUA′, there are
two numbers 0<l <l′ that arenot in the spectrum ofLaplacians suchthat L
are super determinants defined above. The section det(D ) of (detH+ )−1
A (l,l′) ⊗
detH− isinvertible. Bymultiplyingsuchsection,thedeterminantlinebundle
(l,l′)
glues on the overlap.
It is clear that the above construction is compatible with action of Gc, i.e.
L is a line bundle over the stack A/Gc.
When both algebraic and analytic definitions of the determine line bundle
apply, they coincide.
Remark 2.3.
1. The algebraic definition applies to any perfect complexes, in particular
coherent sheaves, over X. The analytic definition only applies for vector
bundles.
2. The analytic definition applies for family (not necessarily bounded) of el-
liptic operators on space of sections of vector bundle. When the elliptic
operator is ∆ for an integrable connection A, it coincides with the al-
A
gebraic definition fiberwise. However, the determinant still makes sense
when F is nonzero while the algebraic definition stops to work.
A
The key observation of the paper is that the analytic definition of determi-
nantbundle isabetter definitiontodiscussKontsevich-Soibelman’sorientation
data. Thisisbecausebyworkingwithnon-integrableconnectionsmosttechnical
difficulties coming from singularities of moduli space are gone.
3 Rational c (L) I
1
Let andL be definedasabove. If is a smoothmanifoldthen firstChern
classMc (L) lies in H2( ,Z). L hasMa topological square root if and only if
1
c (L) is divisible by twMo.
1
Overrational,c (L)canbecomputedbyGrothendieck-Riemann-Rochthe-
1
orem in algebraic context or Atiyah-Singer index theorem in analytic context.
It suffices to check the case when is a compact Riemann surface. In this
M
section, we compute the rational c using GRR theorem,under the assumption
1
that is a smooth proper curve. Proof of the general case, which requires
M
Atiyah-Singer index theorem, will be given in section 5.
Let X be a simply connected CY 3-fold and C be a compact Riemman
surface. Let be the universalsheaf over X C, π be the projectionto C and
E ×
p be the projection to X.
Theorem 3.1. Modulo torsion, the first Chern class of π RHom( , ) is di-
∗
E E
visible by 2.
6
Proof. For simplicity we denote F for RHom( , ). GRR theorem says
E E
ch(πF) td =π (ch(F) td ).
! C ∗ X×C
· ·
By adjunction, we can rewrite it as
ch(πF)=π (ch(F) p∗td ).
! ∗ X
·
The Todd class of X is
c (X)
2
1+ .
12
The Chern character of is
E
c2 2c c3 3c c +3c c4 4c2c +2c2+4c c 4c
ch( )=r+c + 1− 2 + 1− 1 2 3 + 1− 1 2 2 1 3− 4.
1
E 2 6 24
The Chern character of F is
(r 1)c4 4rc2c +2(r+6)c2+4(r 3)c c 4rc
ch(F)=r2+((r 1)c2 2rc )+ − 1− 1 2 2 − 1 3− 4.
− 1− 2 12
Apply GRR, we obtain
(r 1)(c4+c2c (X)) rc c (X) rc2c (r+6)c2 (r 3)c c rc
c (πF)=[ − 1 1 2 2 2 1 2+ 2+ − 1 3 4][X].
1 !
12 − 6 − 3 6 3 − 3 ·
Separate terms depending on rank and terms independent of rank.
c2c (X)+c4 c c (X) c2 c2c c c c c2c (X)+c4
c (πF)=r[ 1 2 1 2 2 + 2 1 2 + 1 3 4] [X] [ 1 2 1 c2+c c ] [X]
1 ! 12 − 6 6 − 3 3 − 3 · − 12 − 2 1 3 ·
c2c (X)+c4
=2rc (π ) [ 1 2 1 c2+c c ] [X]
1 !E − 12 − 2 1 3 ·
(3.1)
The rank depending term is even since c (π ) belongs to H2(C,Z). We
1 !
E
need to show the rank independent term is even.
Lemma 3.2. Let A be a class in H2(X,Z). Then
2A3+A c (X) 0 mod 12.
2
∪ ≡
Proof. Let A be c (D) for some divisor D on X. By GRR,
1
A3 A c (X)
2
χ( (D))= + ∪ .
X
O 6 12
Since this must be an integer, the lemma follows.
By Kunneth formula, c can be written as p∗A+π∗B where A H2(X,Z)
1
and B H2(C,Z). By previous lemma, the term ∈
∈
c2c (X)+c4 2ABc (X)+4A3B
1 2 1 [X]= 2 [X].
12 · 12 ·
7
is even.
Supposerisodd,thereexistsadivisorDsuchthatc ( (D))iseven. There-
1
E
fore, we can assume c c to be even. Finally, c2 is even again by Kunneth
1 3 2
formula.
If is a line bundle, the theorem holds trivially. Let be a sheaf of rank
E E
r =2k >0, we consider a short exact sequence
0 //L // //Q //0
E
where L is a line bundle and Q is a sheaf of rank 2k 1.
−
ch( ∨ )=ch(L∨ L)+ch(Q∨ Q)+ch(L∨ Q)+ch(L Q∨).
E ⊗E ⊗ ⊗ ⊗ ⊗
The sum of the last two terms is even because the odd terms get canceled and
the even terms get doubled. Then the even rank case is proved.
4 Some results in topology and gauge theory
We recall some basics on gauge theory. Our main reference is Chapter 5 of [6].
Let X be a compact complex manifold and G be a compact Lie group. Unless
we specify, G will be taken to be SU(n).
Let P be a principal G bundle over X. Let A = A be the space of
X,P
connections on P and G be the gauge group. The main theme of this section
is the topology of the orbit space B = A/G. Sometime, when there is doubt
about which manifold is involved, we write B instead.
X
Because the action of G is not free, it is much easier to work with framed
connections. If x is a base point on X, a framed connection is a pair (A,φ)
0
whereAisaconnectionandφisanisomorphismofG-spacesφ:G P . The
gauge group G acts naturally on space of framed connections and→we wx0rite B˜
for the space of equivalence classes
B˜ =(A Hom(G,P ))/G.
× x0
Thereis anaturalactionofthefinite dimensionalgaugegroupGonB˜ such
that the quotient stack is B. One way to think of this quotient is to regard a
framing φ as fixed and define G G to be its stabilizer, that is
0
⊂
G = g G g(x )=1 .
0 0
{ ∈ | }
Then B˜ may be described as A/G and the projection B˜ B is simply the
0
→
quotient map for the remainder of the gauge group,
G/G =G.
0 ∼
We recall a standard theorem in algebraic topology.
8
Theorem 4.1. There is a weak homotopy equivalence
B˜ Map∗(X,BG)
X,P P
≃
where Map∗ denotes base-point-preserving maps and Map∗(X,BG) denotes
P
the homotopy class corresponding to the bundle P X.
→
Proof. See proposition 5.1.4 of [6].
Proposition 4.2. When G equals SU(n) for n 0, H2(B˜ ,Z) is torsion
S6
≫
free.
Proof. Because a principal G bundle over S6 is determined by its transition
function over the equator, we have a homotopy equivalence
B˜ Map∗(S5,SU(n))=Ω5SU(n).
S6
≃
Lemma 4.3. Assume k<2n. There is an isomorphism
HN(ΩkSU(n))=HN(ΩkSU(n+1))
∼
for N <2n k.
−
Proof. We first compute the cohomologyof iterated loop spaces of odd spheres
since they are the building blocks of loop spaces of SU(n). When k < 2n+1,
we claim
Z j =0,2n+1 k
Hj(ΩkS2n+1,Z)= − (4.1)
(cid:26) 0 0<j <2n+1 k
−
Thiscanbeprovedbyinductiononk. Whenk =0,thisisclearlyright. Assume
k is even. Consider the fibration
ΩkS2n+1 PΩk−1S2n+1 Ωk−1S2n+1.
→ →
The condition k < 2n + 1 guarantees the base to be connected and simply
connected. By the induction assumption, the bottom row of E2 page of the
Serre spectral sequence looks like
Z ...0... Z ...0...
where the second copy of Z appear in degree 2n+2 k. Since the path space
−
is contractible, the claim follows. The other half of the claim can be check
similarly.
Consider the spectral sequence for the fibration
ΩkSU(n) ΩkSU(n+1) ΩkS2n+1.
→ →
The j-th row of E2 page is zero when j = 2,3,...,2n k. The convergence
−
of spectral sequence gives HN(ΩkSU(n + 1)) = HN(ΩkSU(n)) when N =
∼
0,1,...,2n 1 k.
− −
9
Remark 4.4. The integral cohomology of ΩkS2n+1 is in general not torsion
free. However,for big n the lower degree component will be torsion free.
Recall the stable special unitary group SU is defined to be the direct limit
SU =limSU(n).
−→
In general, the cohomology functor doesn’t commute with direct limit. The
discrepancy is measured by the derived functor lim1. However, by Lemma 4.3
lim1 vanishes. Therefore, for fixed N and k −→
−→
HN(ΩkSU)=HN(ΩkSU(n))
∼
when n 0.
≫
The Bott periodicity theorem says there is a homotopy equivalence
ΩSU BSU.
≃
We are interested in the first and the second cohomology group of Ω5SU(n),
i.e. N = 1,2. By Bott periodicity, HN(Ω5SU(n),Z) = HN(Ω5SU,Z) =
∼ ∼
HN(ΩSU,Z) when n 0. By Bott’s theorem [4], ΩSU is torsion free and
generated by universal≫Chern classes. As a consequence, H1(B˜ ,Z) = 0 and
S6
H2(B˜ ,Z)=Z for G=SU(n) with n 0.
S6
≫
Because B˜ Map∗(S2,BSU(n)) ΩSU(n), its integral cohomology is
S2
≃ ≃
torsion free and generated by the universal Chern classes c ,...,c . This is a
1 n
special case of Proposition 2.20 of [1].
Similarly, B˜ is homotopic to Ω2SU(n). By Lemma 4.3, when n 0
S3
≫
Hi(Ω2SU(n)) = Hi(SU) = 0 for i = 1,2 and H3(Ω2SU(n)) = H3(SU) =
π (S3)=Z.
3
5 Rational c (L) II
1
In this section, we prove theorem 3.1 for not necessarily smooth scheme.
M
Let X be a simply connected CY 3-fold. As we see in section 2, there is a
family of elliptic operators D over the space of framed connections B˜ . The
A X
family has determinant line bundle L.
Denote the index bundle of the elliptic complex
DA
Leven nn -- Lodd
DA∗
by ind(D,E). We are interested in the case of index bundle of the universal
principal bundle over B˜ X.
X
P ×
10