Table Of ContentG –instantons on generalised Kummer
2
constructions. I
Thomas Walpuski
January 26, 2012
2
1
0
Abstract
2
n In this article we introduce a method to construct G2–instantons on G2-
a manifoldsarisingfromJoyce’sgeneralisedKummerconstruction[Joy96]. The
J method is based on gluing anti-self-dual instantons on ALE spaces to flat
6 bundlesonG2-orbifoldsoftheformT7/Γ. Weusethisconstructiontoproduce
2 non-trivial examples of G2–instantons. Finally, we discuss the relevance of
our work to the computation of a conjectural G2 Casson invariant as well as
] potentialfutureapplications.
G
D
1 Introduction
.
h
t
a Theseminalpaper[DT98]ofDonaldson-Thomashasinspiredaconsiderableamount
m
of work related to gauge theory in higher dimensions. Tian [Tia00] and Tao–Tian
[ [TT04] made significant progress on important foundational analytical questions.
Recent work of Donaldson–Segal [DS09] and Haydys [Hay11b,Hay11a] shed some
2
v light on the shape of the theories to be expected.
9 In this article we will focus on the study of gauge theory on G2-manifolds.
0 These are7-manifoldsequippedwith a torsion-freeG -structure. The G -structure
2 2
6
allows to define a special class of connections, called G –instantons. These share
2
6
many formal properties with flat connections on 3–manifolds and it is expected
.
9 thatthereareG –analoguesofthose3–manifoldinvariantsrelatedto“countingflat
2
0
connections”, i.e., the Casson invariant, instanton Floer homology, etc.
1
So far non-trivial examples of G –instantons are rather rare. By exploiting the
1 2
: specialgeometryoftheknownG2–manifoldssomeprogresshasbeenmaderecently.
v
At the time of writing, there are essentially two methods for constructing compact
i
X G –manifolds in the literature. Both yield G –manifolds close to degenerate lim-
2 2
r its. One is Kovalev’s twisted connected sum construction [Kov03], which produces
a
G –manifoldswith“longnecks”fromcertainpairsofCalabi-Yau3-foldswithasymp-
2
totically cylindricalends. A technique for constructingG –instantonsonKovalev’s
2
G –manifolds has recently been introduced by Sa´ Earp [SE11a,SE11b]. The other
2
(and historically the first) method for constructing G –manifolds is due to Joyce
2
and is based on desingularising G –orbifolds [Joy96]. In this article we introduce a
2
method to construct G –instantons on Joyce’s manifolds.
2
To setup the frameworkfor our construction,let us briefly review the geometry
of Joyce’s construction. Give T7 the structure of a flat G –orbifold and let Γ be
2
a finite group of diffeomorphisms of T7 preserving the flat G –structure. Then
2
Y = T7/Γ is a flat G –orbifold. In general, the singular set S of Y can be quite
0 2 0
complicated. For the purpose of this article, we assume that each of the connected
components S of S has a neighbourhood T modelled on (T3×C2/G )/H . Here
j j j j
G isanon-trivialfinitesubgroupofSU(2)andH isafinitesubgroupofisometries
j j
ofT3×C2/G actingfreelyonT3. WecallG –orbifoldsY satisfyingthiscondition
j 2 0
1
admissible. Now, for each j, pick an ALE space X asymptotic to C2/G and an
j j
isometricactionρ ofH onX whichis asymptotic to the actionof H onC2/G .
j j j j j
Given such a collection {(X ,ρ )} of resolution data for Y and a small parameter
j j 0
t>0, one can rather explicitly construct a compact 7–manifold Y together with a
t
G –structurethatisclosetobeingtorsion-free. JoyceprovedthatthisG –structure
2 2
can be perturbed slightly to yield a torsion-free G –structure provided t > 0 is
2
sufficiently small.
We will introduce a notion of gluing data ((E ,θ),{(x ,f )},{(E ,A ,ρ˜ ,m )})
0 j j j j j j
compatible with resolution data {(X ,ρ )} for Y , consisting of a flat connection
j j 0
θ on a G–bundle E over Y and, for each j, a G–bundle E together with an
0 0 j
ASD instanton A framed at infinity as well as various additional data satisfying a
j
number of compatibility conditions. With this piece of notation in place, the main
result of this article is the following.
Theorem 1.1. LetY bean admissible flat G –orbifold, let {(X ,ρ )}beresolution
0 2 j j
data for Y and let ((E ,θ),{(x ,f )},{(E ,A ,ρ˜ ,m )}) be compatible gluing data.
0 0 j j j j j j
Then thereis aconstantκ>0suchthatfor t∈(0,κ)onecan constructaG–bundle
E on Y together with a connection A satisfying
t t t
1
p (E )=− k PD[S ] with k = |F |2
1 t j j j 8π2 Aj
j ZXj
X
and
hw (E ),Σi=hw (E ),Σi
2 t 2 j
for each Σ∈H2(Xj)Hj ⊂H2(Yt). Here [Sj]∈H3(Yt) is thehomology class induced
by the component S of the singular set S.
j
If θ is regular as a G –instanton and each of the A is infinitesimally rigid, then
2 j
for t ∈ (0,κ) there is a small perturbation a ∈ Ω1(Y ,g ) such that A +a is a
t t Et t t
regular G –instanton.
2
TheanalysisinvolvedintheproofofTheorem1.1issimilartounpublishedwork
of Brendle on the Yang–Mills equation in higher dimension [Bre03] and Pacard–
Ritor´e’s work on the Allen–Cahn equation [PR03], however our assumptions lead
to some simplifications. From a geometric perspective our result can be viewed as
a higher dimensional analogue Kronheimer’s work on ASD instantons on Kummer
surfaces [Kro91].
Here is an outline of this article. To benefit the reader Sections 2, 3, 4 and
5 contain some foundational material on G –manifolds and G –instantons as well
2 2
as brief reviewsof Joyce’sgeneralisedKummer constructionand Kronheimer’sand
Nakajima’s work on ASD instantons on ALE spaces. In Section 6 we define the
notion of gluing data and prove the first part of Theorem 1.1. We also introduce
weighted H¨older spaces adapted to the problem at hand and prove that the con-
nection A is a good approximation to a G –instanton. In Section 7 we set up the
t 2
analytical problem underlying the proof of Theorem 1.1 and discuss the properties
ofamodelforthelinearisedproblem. WecompletetheproofofTheorem1.1inSec-
tion 8. A number of examples with G=SO(3) are constructedin Section 9, before
we closeourexpositionwitha shortdiscussionofthe relevanceofourworkto com-
puting a conjectural G Casson invariant and hint at potential future applications
2
in Section 10.
Acknowledgements This article is the outcome of work undertaken by the au-
thor for his PhD thesis at Imperial College London, funded by the EuropeanCom-
mission. I would like to thank my supervisor Simon Donaldson for guidance and
inspiration.
2
2 Review of G –manifolds
2
The Lie group G can be defined as the subgroup of elements of GL(7) fixing the
2
3–form
φ =dx123+dx145+dx167+dx246−dx257−dx347−dx356.
0
Heredxijk isashorthandfordxi∧dxj∧dxk andx ,...,x arestandardcoordinates
1 7
on R7. The particular choice of φ is not important. In fact, any non-degenerate
0
3–form φ is equivalent to φ under a change of coordinates, see e.g. [SW10, Theo-
0
rem 3.2]. Here we say that φ is non-degenerate if for each non-zero vector u ∈ R7
the 2–form i(u)φ on R7/hui is symplectic. G naturally is a subgroup of SO(7).
2
To see this, note that any element of GL(7) that preserves φ also preserves the
0
standard inner product and the standard orientation of R7. This follows from the
identity
i(u)φ ∧i(v)φ ∧φ =6g (u,v)vol .
0 0 0 0 0
Inparticular,everynon-degenerate3–formφona7–dimensionalvectorspacenatu-
rally induces an inner product andan orientation. As anaside we shouldpoint out
herethatnon-degenerate3–formsformoneoftwoopenorbitsofGL(7)inΛ3(R7)∗.
For φ in the other open orbit, the above formula yields an indefinite metric of sig-
nature (3,4). In particular, if we take u=v to be a light-like vector, then i(u)φ is
not a symplectic form on R7/hui.
A G –manifold is 7–manifold equipped with a torsion-free G –structure, i.e., a
2 2
reduction of the structure group of TY from GL(7) to G with vanishing intrinsic
2
torsion. From the above it is clear that a G –structure on Y is equivalent to a
2
3–form φ on Y, which is non-degenerate at every point of Y. The condition to
be torsion-free means that φ is parallel with respect to the Levi–Civita connection
associated with the metric induced by φ. This can also be understood as follows.
Let P ⊂ Ω3(Y) denote the subspace of all non-degenerate 3–forms φ. Then there
is a non-linear map Θ: P →Ω4(Y) defined by
Θ(φ)=∗ φ.
φ
Here ∗ is the Hodge-∗-operator coming from the inner product and orientation
φ
induced by φ. Now the G –structure induced by φ is torsion-free if and only if
2
dφ=0, dΘ(φ)=0.
The proof of this fact is an exercise in linear algebra.
Since G –manifolds have holonomy contained in G which can also be viewed
2 2
as the subgroup of Spin(7) fixing a non-trivial spinor, G -manifolds are spin and
2
admitanon-trivialparallelspinor. Inparticular,theyareRicci-flat. Moreover,G –
2
manifoldnaturallycarryapairofcalibrations: theassociative calibration φandthe
coassociative calibration ψ :=Θ(φ). This should be enough to convince the reader,
that G –manifolds are interesting geometric objects worthy of further study.
2
Simple examples of G –manifolds are T7 with the constant torsion-free G –
2 2
structure φ and products of the form T3×Σ, where Σ is a hyperk¨ahler surface,
0
with the torsion-free G –structure induced by
2
φ =δ ∧δ ∧δ +δ ∧ω +δ ∧ω −δ ∧ω
0 1 2 3 1 1 2 2 3 3
where δ ,δ ,δ are constant orthonormal 1–forms on T3 and ω ,ω ,ω define the
1 2 3 1 2 3
hyperk¨ahler structure on Σ. These examples all have holonomy strictly contained
in G . Compact examples with full holonomy G are much harder to come by. In
2 2
Section4,we will reviewJoyce’sgeneralisedKummer constructionwhichproduced
the first examples of G –manifolds of this kind.
2
3
3 Gauge theory on G –manifolds
2
Let(Y,φ)beaG –manifoldandletE beaG–bundleoverY. Denotebyψ :=Θ(φ)
2
the correspondingcoassociativecalibration. AconnectionA∈A(E) onE is called
a G –instanton if it satisfies the equation
2
F ∧ψ =0.
A
Thisequationisequivalentto∗(F ∧φ)=−F andthusG –instantonsaresolutions
A A 2
oftheYang–Millsequation. Infact,ifY iscompact,thenG -instantonsareabsolute
2
minima of the Yang–Mills functional.
Fromananalyticalpointof view the aboveequationsare notsatisfying because
they are not manifestly elliptic. This issue is resolved by considering instead the
equation
∗(F ∧ψ)+d ξ =0
A A
whereξ ∈Ω0(Y,g ). IfY iscompact(orundercertaindecayassumptions)itfollows
E
from the Bianchi identity and dψ = 0 that d ξ = 0. Thus this it is equivalent to
A
the G –instanton equation. Once gauge invariance is accounted for the equation
2
becomes elliptic. The gauge fixed linearisationat a G –instanton A is given by the
2
operator L : Ω0(Y,g )⊕Ω1(Y,g )→Ω0(Y,g )⊕Ω1(Y,g ) defined by
A E E E E
0 d∗
L = A .
A d ∗(ψ∧d )
A A
(cid:18) (cid:19)
It is easy to see that L is a self-adjointelliptic operator. In particular,the virtual
A
dimension of the moduli space
M(E,φ)={A∈A(E):F ∧ψ =0}/G
A
is zero. It is very tempting to try to define a G Casson invariant by“counting”
2
M =M(E,φ). If every G –instanton A on E is regular, this means that L is an
2 A
isomorphism,then M is a smoothzero-dimensionalmanifold, i.e., a discrete set. If
M was compact, then up to questions of orientation we could indeed count M.
Whether there is a rigorous general definition of a G Casson invariant and
2
whetheritis,infact,invariantunderisotopiestheG –structureisanopenquestion.
2
A brief discussion of parts of this circle of ideas can be found in Donaldson–Segal
[DS09, Section 6].
Simple examples of G –instantons are flat connections and ASD instantons on
2
K3 surfaces Σ pulled back to T3 ×Σ. Moreover, the Levi–Civita connection on
a G –manifolds is a G –instanton. We will construct non-trivial examples using
2 2
Theorem 1.1 in Section 9.
4 Joyce’s generalised Kummer construction
Consider T7 together with a constant 3–form φ defining a G –structure and let
0 2
Γ be a finite group of diffeomorphisms of T7 preserving φ . Then Y = T7/Γ
0 0
naturally is a flat G –orbifold. Denote by S the singular set of Y and denote by
2 0
S ,...,S its connected components. In general, S can be rather complicated. In
1 k
this article we makethe assumptionthat eachS hasa neighbourhoodisometricto
j
a neighbourhood of the singular set of (T3×C2/G )/H . Here G is a non-trivial
j j j
finite subgroup of SU(2) and H is a finite subgroup of isometries of T3×C2/G
j j
acting freely on T3. G –orbifolds Y if this kind will be called admissible.
2 0
4
Let Y be an admissible flat G –orbifold. Then there is a constant ζ > 0 such
0 2
that if we denote by T the set of points at distance less that ζ to S, then T de-
composes into connected components T ,...,T , such that T contains S and is
1 k j j
isometric to (T3×B4/G )/H . On T we can write
ζ j j j
φ =δ ∧δ ∧δ +δ ∧ω +δ ∧ω −δ ∧ω
0 1 2 3 1 1 2 2 3 3
whereδ ,δ ,δ areconstantorthonormal1–formsonT3 andω ,ω ,ω canbe writ-
1 2 3 1 2 3
ten as ω = e1∧e2+e3∧e4, ω = e1∧e3−e2∧e4, ω = e1∧e4+e2∧e3 in an
1 2 3
oriented orthonormal basis (e1,...,e4) for (C2)∗.
We willremovethe singularityalongS byreplacingC2/G withanALE space
j j
asymptotic to C2/G . That is a hyperk¨ahlermanifold (X,ω˜ ,ω˜ ,ω˜ ) together with
j 1 2 3
a continuous map π: X →C2/G inducing a diffeomorphisms from X \π−1(0) to
j
(C2\{0})/G such that
j
∇k(π ω˜ −ω )=O(r−4−k),
∗ i i
as r → ∞ for i = 1,2,3 and k ≥ 0. Here r: C2/G → [0,∞) denotes the radius
j
function. Due to work of Kronheimer [Kro89a,Kro89b] ALE spaces are very well
understood. Here is a summary of his main results.
Theorem 4.1 (Kronheimer). Let G be a non-trivial finite subgroup of SU(2). De-
^
note by X the manifold underlying the crepant resolution C2/G. Then for each
three cohomology classes α ,α ,α ∈H2(X,R) satisfying
1 2 3
(α (Σ),α (Σ),α (Σ))6=0
1 2 3
for each Σ∈H (X,Z) with Σ·Σ=−2 there is a unique ALE hyperk¨ahler structure
2
on X for which the cohomology classes of the K¨ahler forms [ω ] are given by α .
i i
There is always a unique crepant resolution of C2/G. It can be obtained by
^
a sequence of blow-ups. The exceptional divisor E of X = C2/G has irreducible
componentsΣ ,...,Σ . BytheMcKaycorrespondence[McK80],thesecomponents
1 k
form a basis of H (X,Z) and the matrix with coefficients C = −Σ ·Σ is the
2 ij i j
Cartanmatrixassociatedwiththe DynkindiagramcorrespondingtoGintheADE
classification of finite subgroups of SU(2).
Asaconsequenceoftheabove,onecanalwaysfindanALEspaceX asymptotic
j
to C2/G . If H is non-trivial,thenwe also requirean isometricactionρ ofH on
j j j j
X which is asymptotic to the action of H on C2/G . A collection {(X ,ρ )} of
j j j j j
this kind is called resolution data for Y .
0
Denote by π : X →C2/G the resolution map for X . Let π :=tπ :X →
j j j j j,t j j
C2/G and let
j
T˜ :=(T3×π−1(B4/G ))/H and T˜ := T˜ .
j,t j,t ζ j j t j,t
j
[
Then using π we can replace each T in Y by T˜ and thus obtain a compact 7–
j,t j 0 j,t
manifold Y . The Betti numbers of Y canbe readoff easily fromthis construction.
t t
It is easy to see that each of the components S of the singular set S give rise to a
j
homologyclass[S ]∈H (Y )andthateachΣ∈H (X )invariantunderH yieldsa
j 3 t 2 j j
homologyclassΣ∈H (Y ). Moreover,onecanworkoutfundamentalgroupπ (Y ).
2 t 1 t
This is very important, because G –manifolds have full holonomy G if and only if
2 2
their fundamental group is finite.
On T˜ there is a G –structure defined by
j,t 2
φ˜ =δ ∧δ ∧δ +t2δ ∧ω˜ +t2δ ∧ω˜ −t2δ ∧ω˜ .
t 1 2 3 1 1 2 2 3 3
5
If we view Y \ S as a subset of Y , then φ˜ almost matches up with φ in the
0 t t 0
following sense. Introduce a function d : Y → [0,ζ] measuring the distance from
t t
the“exceptionalset”by
|π (y)|, p=[(x,y)]∈T˜
j,t j,t
d (p)=
t (ζ, p∈Yt\T˜t.
Then on sets of the form {x∈T :d (x)≥ǫ}, we have φ˜ −φ =O(t4).
t t t 0
Theorem 4.2 (Joyce). Let Y = T7/Γ be a flat G –orbifold and let {(X ,ρ )} be
0 2 j j
resolution data for Y . Then there is a constant κ>0 such that for t∈(0,κ) there
0
is a torsion-free G –structure φ on the 7-manifold Y constructed above satisfying
2 t t
the following conditions. For α∈(0,1) there is a constant c=c(α)>0 such that
kφt−φ˜tkL∞ +tα[φt−φ˜t]C0,α +tk∇(φt−φ˜t)kL∞ ≤ct3/2
holds on T˜ . Moreover, for every ǫ > 0, there is a constant c = c(α,ǫ) > 0 such
j,t
that
kφ −φ k ≤c(ǫ)t3/2
t 0 C0,α
holds on {x∈Y :d (x)≥ǫ}.
t t
Toprovethis,onefirstwritesdownaclosed3–formψ ∈Ω3(Y )thatinterpolates
t t
between φ˜ and φ in the annulus
t 0
R :={x∈Y :d (x)∈[ζ/4,ζ/2]}.
t t t
This can be done reasonably explicitly using cut-off functions, since
t2(π ) ω˜ =ω +d̺
j,t ∗ i i i
with ∇k̺ = O(r−3−k) for k ≥ 0. If t > 0 is sufficiently small this will induce a
i
G –structure which is torsion-free on the complement of R and has small torsion
2 t
on R . The ansatz φ =ψ +dη leads to the following non-linear equation
t t t t
dΘ(ψ +dη )=0.
t t
The gauge fixed linearisation of this equation is the Laplacian on 2-forms ∆. As t
tendstozero,thepropertiesof∆degeneratebutatthesametimeψ becomescloser
t
andclosertobeingtorsion-free. BycarefulscalingconsiderationsJoyce[Joy96,Part
I] was able to show that the equation can always be solved provided t > 0 is
sufficiently small. We should point out that the estimates we give here are rather
stronger than those given originally by Joyce. The stated estimate (which is still
suboptimal) can be established by a careful analysis of Joyce’s construction or,
alternatively, by applying some of the ideas in this article to the above problem.
The details of this argument will be carried out elsewhere.
5 ASD instantons on ALE spaces
LetΓbeafinitesubgroupofSU(2)andletX beanALEspaceasymptotictoC2/Γ.
Fix a flat connection θ over S3/Γ and let E be a G–bundle over X asymptotic at
infinity to the bundle underlying θ. We consider the space A(E,θ) of connections
A on E that are asymptotic atinfinity to θ (at a certainrate)and the based gauge
6
group G of gauge transformations that are asymptotic at infinity to the identity
0
(again at a certain rate). The space
M(E,θ)={A∈A(E,θ):F+ =0}/G
A 0
is called the moduli space of framed ASD instantons on E asymptotic to θ. It does
notdependonthe precisechoicesaslongasthe rateofdecaytoθ isnotfasterthat
r−3 but still fast enough to ensure that kF k < ∞. If ρ: Γ → G denotes the
A L2
monodromy representation of θ, then the group G ={g ∈G:gρg−1 =ρ} acts on
ρ
M(E,θ). This canbe thoughtofas the residualactionof the groupgaugegroupG
(consisting of gauge transformations with bounded derivative).
Theorem 5.1 (Nakajima [Nak90, Theorem 2.6]). The moduli space M(E,θ) is a
smooth hyperk¨ahler manifold.
Formally, this can be seen as an infinite-dimensional instance of a hyperk¨ahler
reduction [HKLR87]. The space A(E,θ) inherits a hyperk¨ahler structure from X
andtheactionofthebasedgaugegroupG hasahyperk¨ahlermomentmapgivenby
0
µ(A) = F+. To make this rigorous one needs to setup a suitable Kuranishi model
A
for M(E,θ). This can be done using weighted Sobolev spaces as in [Nak90]. The
infinitesimaldeformationtheoryisthengovernedbytheoperatorδ : Ω1(X,g )→
A E
Ω0(X,g )⊕Ω+(X,g ) defined by
E E
δ (a)=(d∗a,d+a).
A A A
Proposition 5.2. Let X be an ALE space, let E be a G–bundle over X and let
A∈A(E) be a finite energy ASD instanton on E. Then the following holds.
1. If a∈kerδ decays to zero at infinity, then a=O(r−3).
A
2. If (ξ,ω)∈kerδ∗ decays to zero at infinity, then (ξ,ω)=0.
A
The last part of this proposition tells us that the deformation theory is always
unobstructed and hence M(E,θ) is a smooth manifold. By the first part the tan-
gent space of M(E,θ) at [A] agrees with the L2 kernel of δ and thus the formal
A
hyperk¨ahler structure is indeed well-defined.
Proof of Proposition 5.2. The proof is based on a refined Kato inequality. Since
δ a = 0, it follows that |d|a|| ≤ γ|∇ a| with a constant γ < 1. To see that, recall
A A
that the Kato inequality hinges on the Cauchy-Schwarz inequality |h∇ a,ai| ≤
A
|∇α||α|. The equation δ a = 0 imposes a linear constraint on ∇ a incompatible
A A
with equality in this estimate unless ∇ a=0. This shows that one can find γ <1,
A
such that a refined Kato inequality holds. Indeed one can easily compute that
γ = 3/4. Now, the Weitzenb¨ock formula implies that for σ =2−1/γ2 =2/3
p (2/σ)∆|a|σ =|a|σ−2(∆|a|2−2(σ−2)|d|a||2)
≤|a|σ−2(∆|a|2+2|∇ a|2)
A
=|a|σ−2ha,∇∗∇ ai
A A
=|a|σ−2(hδ∗δ a,ai+h{R,a},ai+h{F ,a},ai)
A A A
≤O(r−4)|a|σ.
Here {R,.} and {F ,·} denote certain actions of the Riemannian curvature R and
A
the curvature F of A respectively. Since we only care about the behaviour of |a|
A
at infinity, we can view it as function on R4 which satisfies ∆R4|a|σ ≤ O(r−4).
Using the analysis in [Joy00, Section 8.3], we can find f on R4 with f = O(r−2)
and∆R4f =(∆R4|a|σ)+. Now, g =|a|σ−f is a decayingsubharmonicfunctionon
7
R4,soitdecaysatleastliker−2. This followsfromamaximumprincipleargument
using alargemultiple ofthe Green’sfunction asa barrier. Since σ =2/3,itfollows
that |a| decays like r−3. This proves the first part. For the second part, note that
d∗d ξ = 0. By part one d ξ = O(r−3), thus integration by parts yields d ξ = 0,
A A A A
hence ξ =0. Similarly, one shows that ω =0.
The dimension of M(E,θ) can be computed using the following index formula
whichcanbeprovedusingtheAtiyah-Patodi-SingerindextheoremandtheAtiyah-
Bott-Lefschetz formula to compute the contribution from infinity.
Theorem 5.3 (Nakajima [Nak90, Theorem2.7]). Let Γ be a non-trivial finite sub-
group of SU(2) and let θ be a flat connection on a G–bundle over S3/Γ. Let X
be an ALE space asymptotic to C2/Γ, let E be a G–bundle over X asymptotic at
infinity to the bundle supporting θ and let A be a finite energy ASD instanton on E
asymptotic to θ. Then the dimension of the L2–kernel of δ is given by
A
2 χ (g)−dimg
dimkerδ =−2 p (g )+ g .
A 1 E
|Γ| 2−trg
ZX g∈XΓ\{e}
Here p (g ) is the Chern-Weil representative of the first Pontryagin class of E and
1 E
χ is the character of the representation ρ: Γ→G corresponding to θ.
g
ThespaceM(E,θ)isusuallynon-compactandoftenincomplete. Thisisrelated
to instantons wandering off to infinity and bubbling phenomena. For details we
refer the reader to [Nak90].
Let Xˆ be a conformal compactification of X. (The point at infinity will be an
orbifoldsingularity,butthiscausesnotrouble). SinceA∈M(E,θ)hasfiniteenergy
it follows from Uhlenbeck’s removable singularities theorem [Uhl82, Theorem 4.1]
that A extends to Xˆ. Now radial parallel transportyields a frame at infinity for A
in which we can write
A=θ+a with ∇ka=O(r−3−k).
This will turn out to be crucial later on. The reader may find it useful to think of
M(E,θ) as a moduli space of ASD instantons on Xˆ framed at the point at infinity
(although equipped with a non-standardmetric).
ThereisaveryrichexistencetheoryforASDinstantonsonALEspaces. Gocho–
Nakajima[GN92]observedthatforeachrepresentationρ: Γ→U(n)thereisabun-
dleR overX togetherwithanASDinstantonA asymptotictotheflatconnection
ρ ρ
determined by ρ, and if σ is a further representation of Γ, then A = A ⊕A .
ρ⊕σ ρ σ
Kronheimer–Nakajima [KN90] took this as the starting point for an ADHM con-
struction of ASD instantons on ALE spaces. One important consequence of their
work is the following rigidity result.
Lemma 5.4 (Kronheimer–Nakajima [KN90, Lemma 7.1]). Each A is infinitesi-
ρ
mally rigid, i.e., the L2–kernel of δ is trivial.
Aρ
By combining this result applied to the regular representation with the index
formulaKronheimer–Nakajimaderivea geometricversionofthe McKaycorrespon-
dence [KN90, Appendix A]. Let ∆(Γ) denote the Dynkin diagram associated to Γ
in the ADE classification of the finite subgroups of SU(2). Each vertex of ∆(Γ)
correspondstoanon-trivialirreduciblerepresentation. Welabelthesebyρ ,...,ρ
1 k
anddenotetheassociatedbundlesbyR andtheassociatedASDinstantonsbyA .
j j
Theorem5.5(Kronheimer–Nakajima). Theharmonic2–formsc (R )= i trF
1 j 2π Aj
form a basis of L2H2(X)∼=H2(X,R) and satisfy
c (R )∧c (R )=−(C−1)
1 i 1 j ij
ZX
8
where C is the Cartan matrix associated with ∆(Γ). Moreover, there is an isome-
try κ ∈ Aut(H (X,Z),·) such that {c (R )} is dual to {κ[Σ ]}, where Σ are the
2 1 j j j
^
irreducible components of the exceptional divisor E of C2/Γ. If X is isomorphic to
^
C2/Γ as a complex manifold, then κ=id.
This resultsis very usefulfor computing the index ofδ where A is constructed
A
outofASDinstantonsoftheformA (bytakingtensorproducts,directsums,etc.).
ρ
Here is a simple example that we will be used later on.
Proposition 5.6. Let X be an ALE space asymptotic to C2/Z . Denote by
k
R = R and A = A the line bundle and ASD instanton associated with
j ρj j ρj
ρ : Z → U(1) defined by ρ (ℓ) = exp 2πijℓ . For n,m ∈ Z let A be the in-
j k j k k n,m
duced ASD instanton on the SO(3)–bundle E =R⊕(R∗ ⊗R ). Then A
(cid:0) n(cid:1),m n n+m n,m
is infinitesimally rigid, asymptotic at infinity to the flat connection over S3/Z
k
induced by ρ ,
j
1 (k−m)m
|F |2 =
8π2 An,m k
Z
and
w (E )=c (R )−c (R )∈H2(X,Z ).
2 n,m 1 n+m 1 n 2
Proof. To see that A is infinitesimally rigid apply Lemma 5.4 to A ⊕A .
n,m n n+m
The statement about the second Stiefel-Whitney class is obvious.
To compute the energy of A , it is enough to note that the first term in
n,m
the index formula is precisely twice the energy and the second term is given by
−2 –times
k
(cid:0) (cid:1)
k−1
χ (g)−dimg 1−cos(2πmj/k)
g
− = =(k−m)m.
2−trg 1−cos(2πj/k)
g6=e j=1
X X
6 Approximate G –instantons
2
LetY be anadmissibleG –orbifold,let{(X ,ρ )} be resolutiondata for Y . Letκ
0 2 j j 0
be asin Theorem4.2 andfort∈(0,κ)denote by (Y ,φ ) the G manifoldobtained
t t 2
from Joyce’s generalisedKummer construction.
Let θ be a flat connection on a G–bundle E over Y . Then the monodromy
0 0
of θ around S induces a representation µ : π (T ) ∼= (Z3×G )⋊H → G of the
j j 1 j j j
orbifold fundamental group of T . If all µ | are trivial there is a straightforward
j j Gj
way to lift E and θ up to Y . In general, more input is required. A collection
0 t
((E ,θ),{(x ,f )},{(E ,A ,ρ˜ ,m )}) consisting of E and θ as above as well for
0 j j j j j j 0
each j the choice of
• a point x ∈S together with a framing f : (E ) →G of E at x ,
j j j 0 xj 0 j
• aG-bundle E overX togetherwith a framedASD instantonA asymptotic
j j j
at infinity to the flat connection on S3/G induced by the representation
j
µ | ,
j Gj
• a lift ρ˜ of the action ρ of H on X to E and
j j j j j
• a homomorphism m : Z3 →G(E )
j j
9
is called gluing data compatible with the resolutiondata {(X ,ρ )} if the following
j j
compatibility conditions are satisfied:
• The action ρ˜ of H on E preserves A and the induced action on the fibre
j j j j
of E at infinity is given by µ | .
j j Hj
• The action of Z3 on E given by m preserves A and the induced action on
j j j
the fibre of Ej at infinity is given by µj|Z3.
• For all h∈H and g ∈Z3 we have ρ˜ (h)m (g)ρ˜ (h)−1 =m (hgh−1).
j j j j j
We shouldpointoutherethatitisbyfar notalwayspossibletoextendachoice
of (E ,θ) and {(E ,A )} to compatible gluing data. This will become clear from
0 j j
the examples in Section 9. With this notation in place we can prove the first part
of Theorem 1.1.
Proposition 6.1. Let ((E ,θ),{(x ,f )},{(E ,A ,ρ˜ ,m )}) be gluing data com-
0 j j j j j j
patible with {(X ,ρ )}. Then for t∈(0,κ) one can construct a G–bundle E on Y
j j t t
together with a connection A satisfying
t
1
p (E )=− k PD[S ] with k = |F |2
1 t j j j 8π2 Aj
j ZXj
X
and
hw (E ),Σi=hw (E ),Σi
2 t 2 j
for each Σ ∈ H2(Xj)Hj ⊂ H2(Yt). Here [Sj] ∈ H3(Yt) is the cohomology class
induced by the component S of the singular set S.
j
Proof. The choices of ρ˜ and m define a lift of the action of Z3⋊H on R3×X
j j j j
to the pullback of E to R3×X . Passing to the quotient yields a G–bundle over
j j
(T3 × X )/H which we still denoted by E . It follows from the compatibility
j j j
conditions that the pullback of A to R3×X passes to this quotientand defines a
j j
connection on E which we still denote by A .
j j
Fix t∈(0,κ). Let R =R ∩T˜ with R and T˜ defined as in Section 4. By
j,t t j,t t j,t
the compatibility conditions the monodromy of A along S on the fibre at infinity
j j
matches up with the monodromy of θ along E | . Thus, by parallel transport the
0 Sj
identification of (E ) with the fibre at infinity of E extends to an identification
0 xj j
of E | with E | . Patching E and the E via this identification yields the
0 Rj,t j Rj,t 0 j
bundle E .
t
Under the identification of E | with E | , we can write
0 Rj,t j Rj,t
A =θ+a with ∇ka =t2+kO(d−3−k)
j j j t
where d : Y → [0,ζ] is the function measuring the distance from the exceptional
t t
set introduced in Section 4. Fix a smooth non-increasing function χ:[0,ζ]→[0,1]
such that χ(s) = 1 for s ≤ ζ/4 and χ(s) = 0 for s ≥ ζ/2. Set χ := χ◦d . After
t t
cutting off A to θ+χ ·a it can be matched with θ and we obtain the connection
j t j
A on the bundle E .
t t
UsingChern-Weiltheoryonecancomputep (E ). Thestatementaboutw (E )
1 t 2 t
follows from the naturality of Stiefel-Whitney classes.
Ift is small,thenthe connectionA is a closeto being a G –instanton. In order
t 2
to make this precise we introduce weighted H¨older norms. It will become more
transparent over the course of the next two sections that these are well adapted to
the problem at hand. We define weight functions by
w (x):=t+d (x) w (x,y):=min{w (x),w (y)}.
t t t t t
10
