A lower bound on the solutions of Kapustin-Witten equations

0.21 MB·

by  Teng Huang

A lower bound on the solutions of Kapustin-Witten
equations
6
1
0
2 Teng Huang

p
e
S

3
2

]
G Abstract
D
Inthisarticle,weconsidertheKapustin-Wittenequationsonaclosed4-manifold.
.
h Westudy certain analytic properties of solutions to the equations on a closed mani-
t
a fold. The main result is that there exists an L2-lower bound on the extra fields over
m
a closed four-manifold satisfying certain conditions if the connections are not ASD
[ connections.Furthermore,wealsoobtainasimilarresultabouttheVafa-Wittenequa-

6 tions.
v
6
8
9 Keywords.Kapustin-Wittenequations,ASD connections,flat GC-connections
7
0
. 1 Introduction
1
0
6
1 LetX beanoriented4-manifoldwithagivenRiemannianmetricg.Weusethemetricon
:
v X to define the Hodge star operator on Λ•T∗X and then write the bundle of 2-forms as
i
X thedirectsumΩ2T∗X = Ω+ Ω− withΩ+ denotingthebundleofself-dual2-formsand
⊕
r
a with Ω− denoting the bundleof anti-self-dual 2-forms, with respect to this Hodge star. If
ω denotesagiven2-form,thenitsrespectiveself-dualandanti-self-dualpartsaredenoted
byω+ andω−.
Let P be a principal bundle over X with structure group G. Supposing that A is the
connection on P, then we denote by F its curvature 2-form, which is a 2-form on X
A
with values in the bundle associated to P with fiber the Lie algebra of G denoted by g.
We define by d the exterior covariant derivativeon section of Λ•T∗X (P g) with
A G
⊗ ×
respect totheconnectionA.
TheKapustin-Wittenequationsare defined on aRiemannian 4-manifoldgivena prin-
cipal bundle P. For most our considerations, G is taken to be SO(3). The equations
T.Huang:DepartmentofMathematics,UniversityofScienceandTechnologyofChina,Hefei,Anhui
230026,PRChina;e-mail:[email protected]
MathematicsSubjectClassification(2010):53C07;58E15
1
2 TengHuang
requireapair(A,φ)ofconnectionon P and sectionofT∗X (P g) tosatisfy
G
⊗ ×
(F φ φ)+ = 0 and (d φ)− = 0 and d φ = 0. (1.1)
A A A
− ∧ ∗
Witten [23, 24, 25, 26], [9], [11] and also Haydys [10] proposed that certain linear com-
binations of the equations in (1.1) and the version with the self and anti-self dual forms
interchanged should also be considered. The latter are parametrized by τ [0,1] which
∈
can bewrittenas
τ(F φ φ)+ = (1 τ)(d φ)+,
A A
− ∧ −
(1 τ)(F φ φ)− = τ(d φ)−, (1.2)
A A
− − ∧ −
d φ = 0.
A
∗
The τ = 0 version of (1.2) and the τ = 1 version of (1.2) is the version of (1.1) that is
definedonX withitssamemetricbutwithitsorientationreversed.InthecasewhenX is
compact,KapustinandWitten[11]provedthatthesolutionof(1.1)withτ (0,1)exists
∈
onlyinthecasewhenP ghaszerofirstPontrjaginnumber,andifso,thesolutionsare
G
×
such that A + √ 1φ defining a flat PSL(2,C) connection. A nice discussions of there
−
equationscan befoundin [8].
IfX iscompact,and (A,φ)obeys(1.1), then
YMC(A+√−1φ) = Z |(FA −φ∧φ)|2 +|dAφ|2 dvolg
X (cid:0) (cid:1)
= F 2 2 F ,φ φ + φ φ 2 [F ,φ],φ dvol
Z | A| − h A ∧ i | ∧ | −h∗ A i g
X (cid:0) (cid:1)
= F 2 4 F+,φ φ +2 (φ φ)+ 2 dvol
Z | A| − h A ∧ i | ∧ | g
X (cid:0) (cid:1)
= F 2 2 F+ dvol
Z | A| − | A|i g
X (cid:0) (cid:1)
= 4π2p (P so(3)).
1 SO(3)
− ×
where so(3) is the Lie algebra of SO(3) and p (P) is the first Pontrjagin number, the
1
second line we used the fact (2.5). This identity implies, among other things, that there
are no solutions to (1.1) in the case when X is compact and the first Pontrjagin number
is positive. It also implies that (A,φ) solves (1.1) when X is compact and p (P
1 SO(3)
×
so(3)) = 0 ifand onlyifA+√ 1φ defines aflat SL(2,C) connectiononX.
−
In [16], Taubes studied the Uhlenbeck style compactness problem for SL(2,C) con-
nections,includingsolutionstotheaboveequations,onfour-manifolds(seealso[17,18]).
In[13],Tanakaobservedthatequations(1.1)onacompactKa¨hlersurfacearethesameas
Simpson’s equations, and proved that the singular set introduced by Taubes for the case
ofSimpson’sequationshasa structureofaholomorphicsubvariety.
AlowerboundonthesolutionsofKapustin-Wittenequations 3
Wedefinetheconfigurationspaces
:= Ω1(X,g ),
P P
C A ×
′ := Ω2,+(X,g ) Ω2,−(X,g ).
P P
C ×
Wealso definethegauge-equivariantmap
KW : ′,
C → C
KW(A,φ) = F+ (φ φ)+,(d φ)− .
A − ∧ A
(cid:0) (cid:1)
MimickingthesetupofDonaldsontheory,theKW-modulispaceis
M (P,g) := (A,φ) KW(A,φ) = 0 / .
KW P
{ | } G
The moduli space M of all ASD connections can be embedded into M via A
ASD KW
7→
(A,0),hereAisanASDconnection.Inparticular,Taubes[14,15]provedtheexistenceof
ASD connections on certain four-manifolds satisfying extra conditions. For any positive
real constantC R+, wedefinetheC-truncatedmodulispace
∈
MC (P,g) := (A,φ) M (P,g) φ C .
KW { ∈ KW | k kL2(X) ≤ }
In this article, we assume that there is a peculiar circumstance in that one obtains an
L2 lower bound on the extra field φ on a closed, oriented, four-dimension manifold, X
satisfiescertain conditionsiftheconnectionisnotan ASD connection.
Theorem 1.1. (Main Theorem) Let X be a closed, oriented, 4-dimensional Riemannian
manifold with Riemannaian metric g, let P X be a principal G-bundle with G being
→
a compact Lie group with p (P) negative and be such that there exist µ,δ > 0 with the
1
propertythatµ(A) µforallA B (P,g),whereB (P,g) := [A] : F+ < δ
≥ ∈ δ δ { k AkL2(X) }
andµ(A)isasin(4.1).Thereexistapositiveconstant,C,withthefollowingsignificance.
If (A,φ) is an L2 solution of MC (P,g), then A is anti-self-dual with respect to the
1 KW
metricg.
Corollary1.2. Let X bea closed, oriented,4-dimensionRiemannianmanifoldwith Rie-
mannaianmetricg , letP beaprincipalSO(3)bundlewithP so(3)hasnegative
SO(3)
×
first Pontrjagin class over X. Then there is an open dense subset, C(X,p (P)), of the
1
Banach space, C(X), of conformal equivalence classes, [g], of Cr Riemannian metrics
onX (forsomeintegerr 3)withthefollowingsignificance.If[g] C(X,p (P)),then
1
≥ ∈
there exist a positiveconstant, C, with the followingsignificance. Supposethat P and X
obeys oneof thefollowingsetsof conditions:
(1)b+(X) = 0;or
(2)b+(X) > 0 and the second Stiefel-Whitney class, w (P) H2(X;Z/2Z), is non-
2
∈
trivial.
If (A,φ) is an L2 solution of MC (P,g), then A is anti-self-dual with respect to the
1 KW
metricg.
4 TengHuang
The organization of this paper is as follows. In section 2, we first review some esti-
mates of the Kapustin-Witten equations. Thanks to Uhlenbeck’s work, we observe that
A must be a flat connection if φ L2 is sufficiently small, when A+√ 1φ is a flat GC-
k k −
connectionand φ kerd∗. In section3, westudy certain analyticpropertiesofsolutions
∈ A
totheKapustin-Wittenequationsontheclosedmanifolds.Insection4,wegeneralizethe
previousobservationtothecaseofKapustin-Wittenequations.Moreprecisely, weshow
that if X satisfy certain conditions and φ is sufficiently small, there exists a ASD
L2
k k
connectionneargivenconnectionAmeasuredby F+ ,thenwecompletetheproofof
k AkL2
main theorem by the similar way for the case of flat GC-connections. In the last section,
wealsoobtainsimilarresultabout theVafa-Witten equations.
2 Fundamental preliminaries
We shall generally adhere to the now standard gauge-theory conventions and notation of
Donaldson and Kronheimer [2]. Throughout our article, G denotes a compact Lie group
and P a smooth principal G-bundle over a compact Riemannnian manifold X of di-
mension n 2 and endowed with Riemannian metric g. For u Lp(X,g ), where
P
≥ ∈
1 p < and k isan integer,wedenote
≤ ∞
k
u := j u pdvol 1/p, (2.1)
k kLpk,A(X) (cid:0)Xj=0 ZX |∇A | g(cid:1)
where : C∞(X,Ω·(g )) C∞(X,T∗X Ω·(g ))isthecovariantderivativeinduced
A P P
∇ → ⊗
bytheconnection,A,onP andLevi-CivitaconnectiondefinedbytheRiemannianmetric,
g,onT∗X,andallassociatedvectorbundleoverX,and j := ... (repeatedj
∇A ∇A◦ ◦∇A
timesforj 0). TheBanach spaces, Lp (X,Ωl(g )), arethecompletionsofΩl(X,g )
≥ k,A P P
withrespect tothenorms(2.1). Forp = ,wedenote
∞
k
kukL∞k,A(X) := esssup|∇jAu|. (2.2)
X X
j=0
Forp [1, )and nonnegativeintegerk, Banach spacedualitytodefine
∈ ∞
Lp′ (X,Ωl(g )) := Lp (X,Ωl(g )) ∗,
−k,A P k,A P
(cid:0) (cid:1)
wherep′ [1, )is thedual exponentdefined by1/p+1/p′ = 1.
∈ ∞
2.1 Identities for the solutions
ThissectionderivessomebasicidentitiesthatareobeyedbysolutionstoKapustin-Witten
equations.
AlowerboundonthesolutionsofKapustin-Wittenequations 5
Theorem 2.1. (Weitezenbo¨ckformula)
d∗d +d d∗ = ∗ +Ric( )+ [ F , ]onΩ1(X,g ) (2.3)
A A A A ∇A∇A · ∗ ∗ A · P
where Ric istheRiccitensor.
AsasimpleapplicationofWeitezenbo¨ckformula, wehavethefollowingproposition,
Proposition2.2. If(A,φ) isa solutionofKapustin-Wittenequations,then
∗ φ+Ric φ+2 (φ φ)+,φ = 0. (2.4)
∇A∇A ◦ ∗ ∗ ∧
(cid:2) (cid:3)
Proof. From (d φ)− = 0, wehaved φ = d φ.Then wehave
A A A
∗
d d φ = d d φ.
A A A A
∗
Sinced∗ = d , weobtain
A −∗ A∗
d∗d φ = [F ,φ]. (2.5)
A A −∗ A
From (2.3)and (2.5), wecompletestheproofofProposition2.2.
Using a technique of Taubes ([14], p.166) also described in ([12], p.23–24), we com-
binetheWeitzenbo¨ckformulawithMorrey’smean-valueinequalitytodeduceaboundon
φ L∞ in termsof φ L2. First,werecall ameanvalueinequalityas follow:
k k k k
Theorem2.3. ([12]Theorem3.1.2)LetX beasmoothclosedRimeannianmanifold.For
all λ > 0 there are constants C with the following property. Let V X be any real
λ
{ } →
vectorbundleequippedwithametric,A anysmoothmetric-compatibleconnection,
V
∈ A
andσ Ω0(X,V) anysmoothsectionwith satisfiesthepointwiseinequality
∈
σ, ∗ σ λ σ 2.
h ∇A∇A i ≤ | |
Then σ satisfiestheestimate
σ L∞ Cλ σ L2.
k k ≤ k k
Theorem 2.4. Let X be a compact 4-dimensional Riemannian manifold. There exists a
constant,C = C(X), with thefollowing property.Forany principalbundleP X and
→
anyL2 solution(A,φ) totheKapustin-Wittenequations,
1
φ L∞(X) C φ L2(X).
k k ≤ k k
6 TengHuang
Proof. By Theorem 3.7, wemay assumethat(A,φ) issmooth.Form(2.4), in pointwise,
∗ φ,φ = Ric φ,φ +4 (φ φ)+ 2 . (2.6)
h∇A∇A i − h ◦ i | ∧ |
(cid:0) (cid:1)
SinceX iscompact, wegetapointwiseboundoftheform
∗ φ,φ λ φ 2 (2.7)
h∇A∇A i ≤ | |
for a constant λ depending on Riemannian curvature of X. From (2.7) and Theorem 2.3,
weobtain
φ L∞(X) C φ L2(X).
k k ≤ k k
2.2 Flat G -connections
C
Let P X be a principal G-bundle with G being a compact Lie group with p (P) is
1
→
zero, then thesolutions(A,φ)to theKapustin-Wittenequationsare flat GC-connections:
F φ φ = 0 and d φ = 0 and d φ = 0.
A A A
− ∧ ∗
First,wereviewakeyresultduetoUhlenbeckfortheconnectionswithLp-smallcurvature
(2p > n)[20].
Theorem2.5. ([20]Corollary4.3)LetX beaclosed,smoothmanifoldofdimensionn
≥
2andendowedwithaRiemannianmetric,g,andGbeacompactLiegroup,and2p > n.
Thenthereareconstants,ε = ε(n,g,G,p) (0,1]andC = C(n,g,G,p) [1, ),with
∈ ∈ ∞
the following significance. Let A be a Lp connection on a principal G-bundle P over X.
1
If
F ε,
A Lp(X)
k k ≤
then there exist a flat connection, Γ, on P and a gauge transformation g Lp(X) such
∈ 2
that
(1)d∗ g∗(A) Γ = 0onX,
Γ −
(2) g(cid:0)∗(A) Γ L(cid:1)p C FA Lp(X) and
k − k 1,Γ ≤ k k
(3) g∗(A) Γ C F .
k − kLn/2 ≤ k AkLn/2(X)
1,Γ
Theorem 2.6. Let X be a closed, oriented, 4-dimensional Riemannian manifold with
Riemannaian metric g, let P X be a principal G-bundle with G being a compact Lie
→
groupwithp (P) = 0.Thereexistapositiveconstant,C,withthefollowingsignificance.
1
If (A,φ)isan L2 solutionofMC (P,g),then Aisa flatconnection.
1 KW
AlowerboundonthesolutionsofKapustin-Wittenequations 7
Proof. ByTheorem3.7,wecanassume(A,φ)issmooth.FromtheTheorem2.4,wehave
φ L∞(X) C7 φ L2(X),
k k ≤ k k
whereC isonlydependentonmanifold.Since(A,φ)isasolutionoftheKapustin-Witten
7
equations,wehave
F φ φ C φ 2 ,
k AkLp(X) ≤ k ∧ kLp(X) ≤ 8k kL2(X)
wherep > 2.Wecanchoose φ C sufficientlysmallsuchthatC C2 ε,where
L2(X) 8
k k ≤ ≤
ε is the constant in the hypothesisof Theorem 2.5. Then from Theorem 2.5, there exist a
flat connectionΓ such that
A Γ C F .
L2(X) 9 A L2(X)
k − k 1 ≤ k k
UsingtheWeitezenbo¨ck formula,we haveidentities
(d∗d +d d∗)φ = ∗ φ+Ric φ, (2.8)
Γ Γ Γ Γ ∇Γ∇Γ ◦
and
(d∗d +d d∗)φ = ∗ φ+Ric φ+ [ F ,φ], (2.9)
A A A A ∇A∇A ◦ ∗ ∗ A
whichlead to thefollowingtwointegralinequalities
φ 2 + Ric φ,φ 0. (2.10)
k∇Γ kL2(X) Z h ◦ i ≥
X
and
φ 2 + Ric φ,φ +2 F 2 = 0. (2.11)
k∇A kL2(X) Z h ◦ i k Ak
X
On theotherhand,wealso haveanotherintegralinequality
φ φ 2 [A Γ,φ] 2
k∇A −∇Γ kL2(X) ≤ k − kL2(X)
C A Γ 2 φ 2 (2.12)
≤ 10k − kL4(X)k kL4(X)
C F 2 φ 2 .
≤ 11k AkL2(X)k kL2(X)
Combingtheseinequalities,wearriveat
0 φ 2 + Ric φ,φ
≤ k∇Γ kL2(X) Z h ◦ i
X
φ 2 + Ric φ,φ + φ φ 2
≤ k∇A kL2(X) Z h ◦ i k∇A −∇Γ kL2(X)
X
(C φ 2 2) F 2 .
≤ 12k kL2(X) − k AkL2(X)
We can choose φ C sufficiently small such that C C2 1, then F vanishes.
L2(X) 12 A
k k ≤ ≤
8 TengHuang
2.3 A vanishing theorem on extra fields
Asusual, wedefinethestabilizergroupΓ ofAinthegaugegroup by
A P
G
Γ := g g∗(A) = A .
A P
{ ∈ G | }
Definition 2.7. A connection A is said to be irreducible if the stabilizer group Γ is
A
isomorphicto thecentreofG, and Aiscalled reducibleotherwise.
Lemma 2.8. ([2] Lemma 4.3.21) If A is an irreducible SU(2) or SO(3) anti-self-dual
connection on a bundle E over a simply connected four-manifoldX, then the restriction
ofAto anynon-emptyopen setin X is alsoirreducible.
Theorem 2.9. Let X be a simply-connected Riemannian four-manifold, let P X be
→
an SU(2) or SO(3) principalbundle.If A is irreducibleanti-self-dualconnection
P
∈ A
andφ Ω1(X,g )satisfy
P
∈
φ φ = 0 and d φ = 0 and d∗φ = 0
∧ A A
thenφ = 0.
Proof. Since F+ = 0, φ φ = 0, then φ has at most rank one. Let Zc denote the
A ∧
complementofthezeroofφ.Byuniquecontinuationoftheellipticequation(d +d∗)φ =
A A
0,Zc iseitheremptyordense.
The Lie algebra of SU(2) or SO(3) is three-dimensional, with basis σi and
i=1,2,3
{ }
Liebrackets
σi,σj = 2ε σk.
ijk
{ }
In alocal coordinate,wecan set φ = 3 φ σi, whereφ Ω1(X).Then
i=1 i i ∈
P
0 = φ φ = 2(φ φ )σ3 +2(φ φ )σ2 +2(φ φ )σ1.
1 2 3 1 2 3
∧ ∧ ∧ ∧
Wehave
0 = φ φ = φ φ = φ φ . (2.13)
1 2 3 1 2 3
∧ ∧ ∧
On Zc, φ is non-zero, then without loss of generality we can assume that φ is non-zero.
1
From (2.13), thereexistfunctionsµand ν such that
φ = µφ and φ = νφ .
2 1 3 1
Hence,
φ = φ (σ1 +µσ2 +νσ3)
1
σ1 +µσ2 +νσ3
= φ (1+µ2 +ν2)1/2( ).
1
1+µ2 +ν2
p
AlowerboundonthesolutionsofKapustin-Wittenequations 9
Then on Zc write φ = ξ ω for ξ Ω0(Zc,g ) with ξ,ξ = 1, and ω Ω1(Zc). We
P
⊗ ∈ h i ∈
compute
0 = d (ξ ω) = d ξ ω ξ dω,
A A
⊗ ∧ − ⊗
0 = d (ξ ω) = d ξ ω ξ d ω.
A A
∗ ⊗ ∧∗ − ⊗ ∗
Takingtheinnerproductwithξ andusingtheconsequenceof ξ,ξ = 1that ξ,d ξ = 0,
A
h i h i
wegetdω = d∗ω = 0.Itfollowsthatd ξ ω = 0.Sinceω isnowherezeroalongZc,we
A
∧
must have d ξ = 0 along Zc. Therefore, A is reducible along Zc. However according to
A
[2] Lemma 4.3.21, A is irreducible along Zc. This is a contradiction unless Zc is empty.
Therefore Z = X,so φis identicallyzero.
3 Analytic results
3.1 The Kuranishi complex
The most fundamental tool for understanding moduli space of anti-self-dual connection
isthecomplexassociatetoan anti-self-dualconnection A givenby
asd
0 Ω0(g ) dAasd Ω1(g ) d+Aasd Ω2,+(g ) 0.
P P P
→ −−−→ −−−→ →
Thecomplexassociated toKapustin-Wittenequationsistheform
d0 d1
0 Ω1(g ) (A,φ) Ω1(g ) Ω1(g ) (A,φ) Ω2,−(g ) Ω2,+(g ) 0,
P P P P P
→ −−−→ × −−−→ × →
where d1 is the linearization of KW at the configuration (A,φ), and d0 gives the
(A,φ) (A,φ)
action of infinitesimal gauge transformations. These maps d0 and d1 form a com-
(A,φ) (A,φ)
plexwheneverKW(A,φ) = 0.
The action of g on (A,φ) Ω1(g ), and the corresponding infinitesimal
P P P
∈ G ∈ A ×
actionofξ Ω0(g ) is
P
∈
d0 : Ω0(g ) Ω1(g ) Ω1(g ),
(A,φ) P → P × P
d0 (ξ) = ( d ξ,[ξ,φ]).
(A,φ) − A
ThelinearizationofKW at thepoint(A,φ) isgivenby
d1 (a,b) = (d b+[a,φ])−,(d a+[b,φ])+ .
(A,φ) A A
(cid:0) (cid:1)
Wecompute
d1 d0 (ξ) = [ξ,(d φ)−],[ξ,(F +φ φ)+] .
(A,φ) ◦ (A,φ) A A ∧
(cid:0) (cid:1)
10 TengHuang
Thedualcomplexis
d1,∗ d0,∗
0 Ω2,−(g ) Ω2,+(g ) (A,φ) Ω1(g ) Ω1(g ) (A,φ) Ω0(g ) 0.
P P P P P
→ × −−−→ × −−−→ →
Therecodifferentialsare
d1,∗ (a′,b′) = (d∗b′ [φ,a′],d∗a′ + [φ,b′]),
(A,φ) A −∗ A ∗
and
d0,∗ (a,b) = ( d∗a+ [φ, b]).
(A,φ) − A ∗ ∗
Proposition3.1. ThemapKW(A,φ) hasanexact quadraticexpansiongiven by
KW(A+a,φ+b) = KW(A,φ)+d1 (a,b)+ (a,b),(a,b) ,
A,φ { }
where (a,b),(a,b) is thesymmetricquadraticformgivenby
{ }
(a,b),(a,b) := [a,b]−,(a a+[b,b])+ .
{ } ∧
(cid:0) (cid:1)
Givenfixed(A ,φ ),welookforsolutionstotheinhomogeneousequationKW(A +
0 0 0
a,φ +b) = ψ .By Proposition3.1, thisequationis equivalentto
0 0
d1 + (a,b),(a,b) = ψ KW(A ,φ ). (3.1)
A0,φ0 { } 0 − 0 0
To makethisequationelliptic,it’snatureto imposethegauge-fixingcondition
d0,∗ (a,b) = ζ.
(A0,φ0)
Ifwedefine
:= d0,∗ +d1 ,
D(A0,φ0) (A0,φ0) (A0,φ0)
ψ = ψ KW(A ,φ ).
0 0 0
−
thentheellipticsystemcan berewritten as
+ (a,b),(a,b) = (ζ,ψ). (3.2)
D(A0,φ0) { }
Thisissituationis considerin [6]equation3.2in thecontextofPU(2)monopoles.
3.2 Regularity and elliptic estimates
First wesummarizetheresultof [6], which applyverbatimto Kapustin-Wittenequations
uponreplacing thePU(2) spinorΦby φ,