Complex anti-self-dual instantons and Cayley subma

arXiv:math/0302094v2[math.DG]13Aug2003Complexanti-self-dualinstantonsandCayleysubmanifoldsSimonBrendleAugust13,20031IntroductionLetMbeamanifoldofdimension8,andletΩbea4-formwhichdefinesanalmostSpin(7)-structureonM.AnΩ-anti-self-dualinstantonisacon-nectionAonavectorbundleoverMsuchthatthecurvatureFAsatisfiesFA+∗(Ω∧FA)=0.(1)IfMisanalmostCalabi-Yaumanifold,thenthe4-formΩcanbewrittenasΩ=4Re(θ)+12ω2,whereω∈Ω1,1(M)denotesthesymplecticformandθ∈Ω0,4(M)isthecomplexvolumeform.Thecomplexvolumeforminducesananti-linearinvolution∗θ:Ω0,2(M)→Ω0,2(M).Thentheanti-self-dualityequation(1)isequivalenttoF1,1A·ω=0(2)and(1+∗θ)F0,2A=0.(3)Thespaceof2-formssplitsasadirectsumΛ2TM=Λ2+TM⊕Λ2−TM,(4)whereΛ2+TM={ϕ∈Λ2M:3ϕ−∗(Ω∧ϕ)=0}(5)andΛ2−TM={ϕ∈Λ2M:ϕ+∗(Ω∧ϕ)=0}.(6)1NotethatΛ2+Misavectorspaceofdimension7andΛ2−(M)isavectorspaceofdimension21.LetP+andP−betheprojectionsassociatedtothesplitting(4).ThisimpliesP+ϕ=14(ϕ+∗(Ω∧ϕ))andP−ϕ=14(3ϕ−∗(Ω∧ϕ)).WedenotebyΩ2+(M)thespaceofsectionsofthevectorbundleΛ2+TM.Similarly,Ω2−(M)isthespaceofsectionsofthevectorbundleΛ2−TM.IfΩisclosed,thentheanti-self-dualityequation(1)impliestheYang-MillsequationD∗AFA=0.Theequations(1),(2)generalizetheanti-self-dualequationsindimension4(seee.g.[7,23]),andhavebeenstudiedbyvariousauthors,includingS.K.DonaldsonandR.P.Thomas[8,26],L.Baulieu,H.Kanno,andI.M.Singer[3],J.Chen[6],andG.Tian[27].Thesesubmanifoldsarealsoofconsiderableinterestinmathematicalphysics.G.TianconstructedacompactificationofthemodulispaceofΩ-anti-self-dualinstantonsoverM.HeprovedthateverysequenceAkofΩ-anti-self-dualinstantonsoverMhasasubsequence,stilldenotedbyAk,suchthatlimk→∞ZMc2(Ak)∧ψ=ZMc2(A∞)∧ψ+ZSΘψ,wherec2denotesthe4-formrepresentingthesecondChernclassofthebun-dle,andψisasmooth4-formonM.Furthermore,A∞isaΩ-anti-self-dualinstantonwhichissmoothoutsideasetofvanishingH4-measure.Further-more,SisaCayleysubmanifold,i.e.asubmanifoldcalibratedbythe4-formΩ.CayleysubmanifoldswerestudiedbyR.HarveyandH.B.Lawson[9].Thereisarichclassofexamples.Forinstance,thisclasscontainsaslimitingcasestheholomorphicsubvarietiesandthespecialLagrangiansubmanifoldsofM.SpecialLagrangiansubmanifoldshavebeenstudiedextensively,seee.g.[10].Cayleysubmanifoldsplayaroleinhigh-energyphysics,seeforexample[4].Ouraiminthispaperistoconstructsmoothcomplexanti-self-dualinstan-tonssuchthattheenergydensity|FA|2isconcentratednearagivenCayleysubmanifoldS.2Inthefirststep,weconstructasuitablefamilyofapproximatesolutions.Tothisend,weassumethatthenormalbundleNScanbeendowedwithacomplexstructureJandacomplexvolumeformω.Eachapproximatesolutionisdescribedbyaset(v,λ,J,ω),wherevisasectionofthenormalbundleofS,λisapositivefunctiononS,and(J,ω)isaSU(2)-structureonNS.Thecovariantderivativeofthepair(J,ω)canbedescribedbya1-formθwithvaluesintheLiealgebraΛ2+NS.Thecovariantderivativeofthe4-formΩcanbewrittenintheform∇XΩ=8Xk=1iekα(X)∧iekΩ,whereαisa1-formwithvaluesinΛ2+TM.Weconsidertheellipticcomplex0−→Ω0(M)d−→Ω1(M)P+d−→Ω2+(M)−→0.Thefirstandthesecondcohomologygroupsassociatedtothisellipticcom-plesareH0(M)andH1(M).ThethirdcohomologygroupisdenotedbyH2+(M).Theorem1.1.SupposethatH2+(M)=0.Then,foreachε0,thereexistsamappingΞεwhchassignstoeachsetofglueingdata(v,λ,J,ω)∈C2,γ(S)asectionofthevectorbundleV⊕WofclassCγ(S)suchthatthefollowingholds.(i)If(v,λ,J,ω)isasetofglueingdatasuchthatkvkC1,γ(S)≤K,kλkC1,γ(S)≤K,infλ≥1,k(J,ω)kC1,γ(S)≤K,thenwehavetheestimateΞε(v,λ,J,ω)−4projV4Xi,j=1(∇ivk+αik,lvl)ei⊗e⊥k,projW4Xi,k,l=1(λ−1∇iλδkl+θi,kl+αik,l)ei⊗e⊥k⊗e⊥lCγ(S)≤Cε132.3(ii)IfΞε(v,λ,J,ω)=0,thentheapproximatesolutionAcorrespondingto(v,λ,J,ω)canbedeformedtoanearbyconnection˜AsatisfyingF˜A+∗(Ω∧F˜A)=0.InSection2,westudythemappingpropertiesofamodeloperatoronR8.InSection3,weconstructafamilyofapproximatesolutionsoftheYang-Millsequations.Moreprecisely,givenanysetofglueingdata(v,λ,J,ω)satisfyingkvkC1,γ(S)≤K,kλkC1,γ(S)≤K,infλ≥1,k(J,ω)kC1,γ(S)≤K,weconstructaconnectionAsuchthatkFA+∗(Ω∧FA)kCγ3(M)≤Cε2.Here,theweightedH¨olderspaceCγν(M)isdefinedaskukCγν(M)=sup(ε+dist(p,S))ν|u(p)|+sup4dist(p1,p2)≤ε+dist(p1,S)+dist(p2,S)(ε+dist(p1,S)+dist(p2,S))ν+γ|u(p1)−u(p2)|dist(p1,p2)γ.InSection4,wederiveestimatesforthelinearizedoperatorwhichareinde-pendentofε.InSection5,weapplythecontractionmappingprincipletodeformtheapproximatesolutionAtoanearbyconnection˜A=A+asuchthat(I−P)(F˜A+∗(Ω∧F˜A))=0,where(I−P)isthefibrewiseprojectionfromCγν(M)tothesubspaceGγν(M).Inparticular,ifthebalancingconditionP(F˜A+∗(Ω∧F˜A))=0issatisfied,then˜AisanΩ-anti-self-dualinstanton.InSection6,wecalculatetheleadingtermintheasymptoticexpansionofP(F˜A+∗(Ω∧F˜A))=0.ThisconcludestheproofofTheorem1.1.TheauthorisgratefultoProfessorGerhardHuiskenandProfessorGangTianfordiscussions.42ThemodelproblemonR8TheSpin(7)-structureonR8isgivenbyΩ=−e1∧e2∧e⊥1∧e⊥2−e1∧e2∧e⊥3∧e⊥4−e3∧e4∧e⊥1∧e⊥2−e3∧e4∧e⊥3∧e⊥4+e1∧e3∧e⊥2∧e⊥4−e1∧e3∧e⊥1∧e⊥3−e2∧e4∧e⊥2∧e⊥4+e2∧e4∧e⊥1∧e⊥3−e1∧e4∧e⊥2∧e⊥3−e1∧e4∧e⊥1∧e⊥4−e2∧e3∧e⊥2∧e⊥3−e2∧e3∧e⊥1∧e⊥4+e1∧e2∧e3∧e4+e⊥1∧e⊥2∧e⊥3∧e⊥4.Hence,the2-formse1∧e2+e3∧e4−e⊥1∧e⊥2−e⊥3∧e⊥4,e1∧e3−e2∧e4−e⊥1∧e⊥3+e⊥2∧e⊥4,e1∧e4+e2∧e3−e⊥1∧e⊥4−e⊥2∧e⊥3,e1∧e⊥1+e2∧e⊥2+e3∧e⊥3+e4∧e⊥4,e1∧e⊥2−e2∧e⊥1−e3∧e⊥4+e4∧e⊥3