The space of the Donaldson-Thomas instantons on compact Kahler threefolds, I

compactK¨ahlerthreefolds,I.

YuujiTanaka

Abstract

Inthisarticle,weintroduceperturbedHermitian-Einsteinequa-tions,whichwecalltheDonaldson-Thomasequations,andstudylocal

propertiesofthemodulispaceofsolutionstotheequationssuchastheinfinitesimaldeformationandtheKuranishimapofthemodulispace.

1Introduction

Inthelate90’s,S.K.DonaldsonandR.P.Thomasproposedgaugethe-oryinhigherdimensionsin[DT].Subsequently,R.P.ThomasintroducedtheholomorphicCassoninvariantin[Th].TheholomorphicCassoninvari-antisadeformationinvariantofasmoothprojectiveCalabi-Yauthreefold,whichisobtainedfromthemodulispaceofsemi-stablesheaves.ThiscanbeviewedasacomplexanalogyoftheTaubes-CassoninvariantdefinedbyC.H.Taubesin[Ta].However,itisnotachievedasyettoconstructtheinvariantina“gaugetheoretic”,orananalyticway.

ThefirstcandidateofgaugeconfigurationswhichwoulddescribethisinvariantcouldbetheHermitian-Einsteinconnection,sincethetheHitchin-Kobayashicorrespondenceinsiststhatthereisaone-to-onecorrespondencebetweentheexistenceoftheHermitian-EinsteinconnectionandtheMumford-TakemotostabilityofanirreduciblevectorbundleoveracompactK¨ahlermanifold(seee.g.[Ko]).However,theHermitian-Einsteinequationsdonotformanellipticsystemevenaftergaugefixingincomplexdimensionthreeandmore,sothismightcausealittleproblem.

Toworkoutthisissue,DonaldsonandThomassuggestedaperturbationoftheHermitian-Einsteinequationsin[Th].ThisperturbationwasalsobroughtinbyBaulieu-Kanno-Singer[BKS]andIqbal-Nekrasov-Okounkov-Vafa[INOV]indifferentcontexts.

1

Letusintroducetheperturbedequationsmentionedabove:LetYbeacompactK¨ahlerthreefoldwiththeK¨ahlerformω,andE=(E,h)ahermi-tianvectorbundleofrankroverY.WeconsiderthefollowingequationsforaconnectionAofE,whichpreservesthehermitianstructureofE,andanEnd(E)-valued(0,3)-formuonY:

F0,2¯∗A+∂A

u=0,F1,1

A∧ω2+[u,u¯]=λ(E)IEω3,

whereλ(E)isaconstantdefinedby

λ(E):=

3(c1(E)·[ω]2)

arelationbetweentheDonaldson-Thomasequationsandthecomplexanti-self-dualequations.

2.1TheDonaldson-Thomasprogram

S.K.DonaldsonandR.P.Thomasproposedhigher-dimensionalanalogueoflow-dimensionalgaugetheoryin[DT].Morespecifically,theysuggestedthefollowingtwodirections;

1.Gaugetheoryincomplex2,3,4dimensions,

2.GaugetheoryonG2-manifoldsandSpin(7)-manifolds.

Thefirstoneconcernsa“complexification”ofthelow-dimensionalgaugetheory.Inthisdirection,ThomasintroducedtheholomorphicCassonin-variantin[Th],whichweshallrefertointhenextsection.Thesecondonecouldberelatedto“topologicalM-theory”proposedin[N],[DGNV].

2.2

TheholomorphicCassoninvariantonCalabi-Yauthree-folds

R.P.ThomasdefinedtheholomorphicCassoninvariantoveraCalabi-Yauthreefoldin[Th].WecansaythatitisacomplexanalogyoftheTaubes-CassoninvariantdefinedbyC.H.Taubesin[Ta].

TheTaubes-Cassoninvariantwasdefinedby“counting”flatconnec-tionsoveracompactorientedrealthree-manifold.DonaldsonandThomasspeculatedthatacomplexanalogyofthisCasson-Taubesinvariantwouldbeobtainedby“counting”holomorphicvectorbundlesoveraCalabi-Yauthreefold.

Basedonthisidea,ThomasdefinedtheholomorphicCassoninvariantofaCalabi-Yauthreefoldbyconstructingavirtualmodulicycleofthemodulispaceofsemi-stablesheavesundersometechnicalconditions.There,heusedwell-developedmachineriessuchasthecompactificationofthemodulispaceofsemi-stablesheavesbyGieseker,andtheperfecttangent-obstructiontheorybyLi-Tian[LT].

2.3

TheHermitian-EinsteinconnectionoverK¨ahlermani-folds

Ontheotherhand,astablevectorbundlehasadifferential-geometricde-scriptionthroughtheHitchin-Kobayashicorrespondence,whichsaysthat

3

thereisaone-to-onecorrespondencebetweentheexistenceoftheHermitian-EinsteinconnectionandtheMumford-TakemotostabilityofanirreducibleholomorphicvectorbundleoveracompactK¨ahlermanifold.

Thus,onemightexpectapossibilityofananalyticdefinitionoftheholomorphicCassoninvariantbyusingthemodulispaceoftheHermitian-Einsteinconnections.

However,thiswouldnotbeliterallytrue.OnecanseeitbyconsideringtheinfinitesimaldeformationoftheHermitian-Einsteinconnection,whichwasstudiedbyH-J.Kim[Ki].Webrieflylookbackonithere.

LetXbeacompactK¨ahlermanifoldofcomplexdimensionnwiththeK¨ahlerformω,E=(E,h)ahermitianvectorbundleoverX.AmetricpreservingconnectionAofEissaidtobeaHermitian-EinsteinconnectionifAsatisfiesthefollowingequations:

F0,2A=0,(2.1)ΛFA

1,1

=λ(E)IE,(2.2)

whereλ(E)isdefinedby

λ(E):=

n(c1(E)·[ω]n−1)

and

′+¯A¯A◦P0,2,:=P+◦DA,D:=DDA

whereP+,P0,2arerespectivelytheorthogonalprojectionsfromΩ2toΩ+,Ω0,2.

H-J.Kimprovedthat(2.4)isanellipticcomplexifAisaHermitian-Einsteinconnection.However,itisobviouslynottheAtiyah-Hitchin-Singertypecomplex[AHS]ifn≥3,sincethereareadditionaltermssuchasΩ0,3(X,End(E))andsoon.

2.4TheDonaldson-ThomasinstantononK¨ahlerthreefolds

Aswesawintheprevioussection,theHermitian-EinsteinconnectionwouldnotworkforananalyticconstructionoftheholomorphicCassoninvariant.Inordertoworkouttheaboveissue,we“perturb”theHermitian-Einsteinequations.ThisperturbationwasintroducedbyS.K.DonaldsonandR.P.Thomas,andtheypointedoutthattheseequationscouldworkforanana-lyticdefinitionoftheholomorphicCassoninvariantin[Th].

LetYbeacompactK¨ahlerthreefoldwiththeK¨ahlerformω,andEahermitianvectorbundleofrankroverY.WeconsiderthefollowingequationsforaconnectionAofE,whichpreservesthehermitianstructureofE,andanEnd(E)-valued(0,3)-formuonY:

0,2¯∗u=0,+∂FAA

1,1

∧ω2+[u,u¯]=λ(E)IEω3.FA

(2.7)(2.8)

Wecalltheseequations(2.7),(2.8)theDonaldson-Thomasequations,and

asolution(A,u)totheseequationsDonaldson-Thomasinstanton.

OnemaythinkoftheseequationsastheHermitian-Einsteinequationswithaperturbationu.However,wethinkofuasaHiggsfield,namely,anewvariable.OneofadvantagesofintroducingthenewfielduisthattheDonaldson-Thomasequationsformanellipticsystemafterfixingagaugetransformation,despitethefactthattheHermitian-EinsteinequationsoncompactK¨ahlerthreefoldsdonotformitinthesameway.

Theseequations(2.7),(2.8)werealsostudiedinphysicssuchas[BKS].Inthatcontext,theseequationsareinterpretedasabosonicpartofdimen-sionalreductionequationsoftheN=1superYang-Millsequationin10dimensionsto6dimensions(seealso[INOV],[NOV]).

¯AF0,2=0,theDonaldson-Remark2.1.IfoneusestheBianchiidentity∂A

5

Thomasequations(2.7),(2.8)arerewrittenby

∂¯∗A

u=0,(2.9)F0,2A=0,

(2.10)F1,1

A∧ω2+[u,u¯]=λ(E)IEω3,

(2.11)

since∂¯A∂¯∗Au=0implies∂¯∗A

u=0oncompactK¨ahlermanifolds.2.5

Thecomplexanti-self-dualequationsandtheDonaldson-Thomasequations

Inthissection,wedescribearelationbetweentheDonaldson-Thomasequa-tionsandthecomplexanti-self-dualequations.

First,weintroducethecomplexanti-self-dualequationsonCalabi-Yaufourfolds:LetXbeacompactCalabi-YaufourfoldwiththeK¨ahlerformωandtheholomorphic(4,0)-formθ.Weassumethefollowingnormalizationconditiononωandθ:

θ∧θ¯=

ω4WechoosetheCayleycalibrationonCalabi-YaufourfoldsasΩ,namely,weput

1

Ω:=4Re(θ)+

3

LocalstructuresofthespaceofDonaldson-Thomasinstantons

Inthissection,westudytheinfinitesimaldeformationandtheKuranishimapofthemodulispaceofDonaldson-Thomasinstantons.

3.1ThespaceofDonaldson-Thomasinstantons

LetYbeaK¨ahlerthreefoldwiththeK¨ahlerformω,E=(E,h)ahermitianvectorbundleoverY.

WedenotebyA(E)=A(E,h)thesetofconnectionsofEwhichpreservethehermitianstructureofE=(E,h).Put

Ω0,3(Y,End(E)):=A0,3(Y)⊗RΩ0(Y,End(E)),

whereA0,3(Y)isthespaceofreal(0,3)-formsoverYandEnd(E)=End(E,h)isthebundleofskew-hermitianendmorphismsofE,and

C(E):=A(E)×Ω0,3(Y,End(E)).

WedenotebyG(E)thegaugegroup,wheretheactionofthegaugegroupisdefinedintheusualway.ThesespacesareallequippedwithC∞-topology.SincetheactionofG(E)onC(E)isproper,C(E)/G(E)isHausdorff.

WedenotebyD(E)thesetofallDonaldson-ThomasinstantonsofE,namely,

D(E):={(A,u)∈C(E);(A,u)satisfiesequations(2.7),(2.8)}.Since(2.7),(2.8)aregaugeinvariant,

D(E)/G(E)→C(E)/G(E)

isinjective.Thus,D(E)/G(E)isalsoHausdorff.Wecall

M(E):=D(E)/G(E)

themodulispaceoftheDonaldson-Thomasinstantons.

3.2Theinfinitesimaldeformation

TheinfinitesimaldeformationofaDonaldson-Thomasinstanton(A,u)isgivenby

0→Ω0−→Ω1⊕Ω0,3−−→Ω+→0,

8

D

D+

(3.1)

where

Ωk:=Ωk(Y,End(E)):=Ak(Y,End(E)),

Ωp,q:=Ωp,q(Y,End(E)):=Ap,q(Y)⊗RΩ0(Y,End(E)),

(3.2)

andAk(Y,End(E))isthespaceoftherealk-formswithvaluesinEnd(E),Ap,q(Y)isthespaceofreal(p,q)-formsoverY,

Ω+:=Ω2∩(Ω2,0⊕Ω0,2⊕Ω0ω)

={φ+φ

¯+fω|φ∈Ω2,0,f∈Ω0},(3.3)

andDandD+aremapsdefinedby

D:g→(DAg,[u,g]),

(3.4)D+:(α,υ)→D+Aα+Λ2([u,υ¯]+[υ,u¯])+D¯∗A

υ(3.5)

forg∈Ω0and(α,υ)∈Ω1⊕Ω0,3,whereΛ:=(∧ω)∗.

If(A,u)istheDonaldson-Thomasinstanton,then(3.1)isacomplex.Infact,D+◦D=0followsdirectlyfromtheequations(2.7),(2.8).Proposition3.1.If(A,u)∈D(E),thenthecomplex(3.1)iselliptic.proof.Weprovethatthecomplex(3.1)isexactatthesymbollevel.

Letw=0bearealcotangentvectoraty∈Y.Wewrite

w=w1,0+w0,1,

wherew1,0isa(1,0)-formandw0,1=

TheexactnessatΩ+justfollowsfromthevanishingofthealternatingsum:

dimΩ0y−dim(Ω1⊕Ω0,3)y+dimΩ+

y

=1−(6+2)+󰀆

2·

3·2

c1(E)2−rc2(E)+r

2

2

󰀇󰀂

3(−1)idimH0,i(Y).i=0

Remark3.2.IfahermitianvectorbundleEadmitsaHermitian-Einstein

connection,then

󰀄

ω∧󰀆rc2(E)−

r−1

M

whereB(u,υ):=Λ2([u,υ¯]+[υ,u¯])andΛ(α):=(∧α)∗.Weput

S(A,u):={(α,υ)∈Ω1⊕Ω0,3|(α,υ)satisfies(3.10),(3.11),(3.12)}.Thereisanaturalmap

P:S(A,u)→D(E)/G(E)

(3.13)

definedby(α,υ)→[(A+α,u+υ)].

WedenotebyEnd0(E):=End0(E,h)thebundleoftrace-freeskew-hermitianendmorphismsofE,anddefine

˜k:=Ωk(Y,End0(E)):=Ak(Y,End0(E)),Ω

˜p,q:=Ωp,q(Y,End0(E)):=Ap,q(Y)⊗RΩ˜0,Ω

˜+:=Ω˜2∩(Ω˜2,0⊕Ω˜0,2⊕Ω˜0ω),Ω

andconsiderthesubcomplexof(3.1):

DD˜1˜+→0.˜0,3−˜0−−→Ω→Ω⊕Ω0→Ω

+

(3.14)

(3.15)

0i˜iWedenotebyH=H(Y,End(E))thei-thcohomologyofthecom-(A,u)(A,u)

plex(3.15)fori=0,1,2.

WedenotebyΓ(A,u)thestabilizerat(A,u)∈D(E)ofthegaugegroupG(E),namely,

Γ(A,u):={g∈G(E)|g(A,u)=(A,u)}.

Wecall(A,u)∈D(E)regularifthestabilizerΓ(A,u)istheidentity,irre-ducibleif(A,u)isregularorΓ(A,u)coincideswiththecenterofthestruc-ˆ(E)thesetofallturegroupofE,andreducibleotherwise.WedenotebyD

irreducibleDonaldson-ThomasinstantonsofE.

ˆ(E).Then,thereexistsaneighborhoodProposition3.3.Let(A,u)∈D

U(A,u)of0inS(A,u)suchthatthemapPinducesahomeomorphismbetweenU(A,u)/Γ(A,u)andaneighborhoodof[(A,u)]inD(E)/G(E).proof.First,weprovethatthemapislocallysurjective:

ˆ(E)iscloseenoughto(A,u)∈Lemma3.4.Supposethat(A+α,u+υ)∈D

ˆ(E),where(α,υ)∈Ω1⊕Ω0,3.ThenthereexistsagaugetransformationD

gwhichsatisfies

(g(A+α)−A,g(u+υ)−υ)∈S(A,u).

11

proof.Sincetheequations(3.10)and(3.11)aregaugeinvariant,whatweneedtoshowis

D∗(g(A+α)−A,g(u+υ)−υ)=0.

Fromtheassumptionthat(A,u)isirreducible,thekerneloftheLaplacian∆(0)onΩ0(Y,End(E))is1-dimensional.WedenotebyStheorthogonalcomponentofH0(Y,End(ES:=󰀃))inΩ0(s∈Ω0;

󰀄

Y,End(E)),namely,

Tr(s)ω3=0󰀅,Y

sothat

Ω0=H0(Y,End(E))⊕S.

NotethatSisanidealoftheLiealgebraΩ0sinceTr([v1,v2])=0forall

v1,v2∈Ω0.

Wedefineamap

F:S×(Ω1⊕Ω0,3)→Sby

F(s,(α,υ)):=D∗(es(A+α)−A,es(u+υ)−u).

(3.16)ThemapFcanbeextendedtoasmoothmap

F:Lpp(Ω⊕Ω)→Lpk+1(S)⊕Lk10,3

k−1(S)

(3.17)

fork>6/p.Now,

∂F

Next,weprovethelocalinjectivityofthemapP:

Lemma3.5.If(A+α1,u+υ1)and(A+α2,u+υ2)aresufficientlycloseto(A,u)andifthereexistsagaugetransformationgsuchthat

g(A+α1,u+υ1)=(A+α2,u+υ2),

(3.20)

then

(α1,υ1)=(α2,υ2).

proof.Ifgissufficientlyclosetotheidentity,namely,g=eswiths∈Ω0closeenoughto0,thenLemma3.5followsfromtheimplicitfunctiontheorem.Infact,since(α1,υ1),(α2,υ2)∈S(A,u),wehave

F(s,(α1,υ1))=D∗(α2,υ2)=0,F(s,(α1,υ1))=D∗(α1,υ1)=0.

Thus,s=0,and(α1,υ1)=(α2,υ2).

Next,weconsiderthecasethatgisanarbitrarygaugetransformation.ByusingthedecompositionΩ0=H0⊕S,wewrite

s=csIdE+s1,

wherecsisaconstant,ands1∈S.Thenthereexistsaconstantz1>0suchthat

||DAs1||Lpk

≥z1||s1||Lpk+1

,(3.21)

Ontheotherhand,from(3.20),wehave

DAs=s◦α2−α1◦s.

(3.22)

Thus,

||DAs1||Lpk

=||DAs||Lp

k

=z2||s||Lpk+1

(||α1||Lpk

+||α2||Lpk

)

(3.23)

=z3(cs||IdE||Lp

k+1

+||s1||Lp+1

)(||α1||Lpk

+||α2||Lpkk

),

wherez2,z3>0aresomeconstants.Hence,weobtain

||scs||IdE||Lp(||α1||Lp+||α2||Lp1||k+1

k

k

)

Lp

k+1

≤

wherez4>0isaconstant.Fromthis,ifα1andα2aresmallenough,thencs=0.Thus,

1

s1cs

isclosetoIdE.Hence,wecanapplytheargumentbefore,wheregissuffi-cientlysmall.

WeintroducetheKuranishimap.Let∆(i),G(i),H(i)(i=0,1,2)beLaplacian,Greenfunction,andharmonicprojectionofthecomplex(3.1).Thesesatisfy

Id=H(i)+∆(i)◦G(i)fori=0,1,2.NotethatLaplacian∆(2)onΩ+iswrittenby

+∗+¯∗D¯′)+D(DA∆(2)=DAAA.

¯′:=DA◦P0,2.Fromthese,theleft-hand-sideof(3.10)becomeswhereDA

+

α+P+(α∧α)+B(u,υ)DA

+∗+

)◦G(2)◦(P+(α∧α)+B(u,υ)))(α+(DA=DA

¯∗D¯′◦G(2)◦(P+(α∧α)+B(u,υ))+D

A

A

(3.25)

+H(2)◦(P+(α∧α)+B(u,υ)).

Thus,weobtaintheorthogonaldecompositionof(3.10):

+∗+

)◦G(2)◦(P+(α∧α)+B(u,υ)))=0,(α+(DADA

¯∗D¯′◦G(2)◦(P+(α∧α)+B(u,υ))=0,D

A

(3.26)(3.27)(3.28)

H

A

(2)

◦(P+(α∧α)+B(u,υ))=0.

Alsotheleft-hand-sideof(3.11)is

∗∗′¯A¯A¯ADυ+Λ(α)(u+υ)=D(υ+D◦G(2)(Λ(α)(u+υ)))

+∗+

)◦G(2)◦(Λ(α)(u+υ))(DA+DA

(3.29)

+H(2)(Λ(α)(u+υ)).

Thus,theorthogonaldecompositionof(3.11)becomes

∗′¯A¯AD(υ+D◦G(2)(Λ(α)(u+υ)))=0,+∗+

)◦G(2)◦(Λ(α)(u+υ))=0,(DADA

(3.30)(3.31)(3.32)

H(2)(Λ(α)(u+υ))=0.

14

Now,wedefinetheKuranishimap

K:Ω1⊕Ω0,3→Ω1⊕Ω0,3

byK(α,υ)

+∗¯′◦G(2)(Λ(α)(u+υ))).)◦G(2)◦(P+(α∧α)+B(u,υ)),υ+D:=(α+(DAA

Weput

i(i)Hi:=Hi(A,u)(Y,End(E)):={ϕ∈Ω|∆ϕ=0}

(3.33)

fori=0,1,2.Then,fromthedefinitionoftheKuranishimap,weobtain

K(S(A,u))⊂H1⊕H0,3,

whereH0,3:=H0,3(Y,End(E))istheharmonicpartofΩ0,3(Y,End(E)).Theorem3.6.Let(A,u)beanirreducibleDonaldson-Thomasinstanton.˜2SupposethatH(A,u)=0.ThenthemapKgivesahomeomorphismfroma

neighborhoodof0inS(A,u)toaneighborhoodof0inH1⊕H0,3.proof.WeextendthemapKtoamap

p10,310,3

K:Lp(Ω⊕Ω)→Lkk(Ω⊕Ω)

for16/p.SincetheFrechetdifferentialofKat0

isidentity,theinversemappingtheoremontheBanachspaceinsiststhatthereexistaneighborhoodU⊂H1⊕H0,3andamap

10,3

L:U→Lpk(Ω⊕Ω)

suchthat

K(L(b,w))=(b,w).

02˜2FromtheassumptionthatH(A,u)=H(A,u)(Y,End(E))=0,wehave

H(2)◦(P+(α∧α)+B(u,υ))=0.

Lemma3.7.If(b,w)∈H1⊕H0,3issmallenough,thenL(b,w)∈S(A,u).

15

proof.Let(α,υ):=L(b,w).Then

b=α+(D+A

)∗

◦G(2)◦(P+(α∧α)+B(u,υ)),w=υ+D¯′A◦G(2)(Λ(α)(u+υ)).

Put

ξ:=D¯∗D¯′AA◦G(2)◦(P+(α∧α)+B(u,υ)),η:=D+

A

(D+A)∗◦G(2)◦(Λ(α)(u+υ)).Since(b,w)∈H1⊕H0,3issmallenough,L(b,w)∈S(A,u)and

H(2)◦(P+(α∧α)+B(u,υ)))=0.

Thus,weobtain

ξ=D+

A

α+P+(α∧α)+B(u,υ),η=D

¯∗A

υ+Λ(α)(u+υ).From(3.34),wehave

ξ0,2=(D+

A

α+P+(α∧α)+B(u,υ))0,2=D

¯Aα0,1+α0,1∧α0,1.Thus,[ξ0,2,α0,1]=[D

¯Aα0,1,α0,1],andξ=G(2)D¯∗AD¯′A(P+(α∧α)+B(u,v))=G(2)D

¯∗AD¯A(α0,1∧α0,1)=G(2)D¯∗A[D¯Aα0,1,α0,1]=

G(2)D¯∗A

[ξ0,2,α0,1]Therefore,

||ξ||Lpk

≤||ξ||Lp

k+1

≤z5||ξ||Lpk

||α||Lpk

,

wherez5>0isaconstant.Thus,ifαissmallenough,ξ=0.

Inasimilarway,weobtain

η=G(2)◦D+

A

(D+∗A)(∗[α,∗(u+υ)])=G(2)D+

A

∗D+A[α0,1,∗(u+υ)]=G(2)D+A

∗([D

¯Aα0,1,∗(u+υ)]−[α0,1,∗D∗Aυ])=

G(2)D+

A

∗[α0,1,∗η].

16

(3.34)(3.35)

(3.36)

(3.37)

(3.38)

(3.39)

Therefore,

||η||Lp≤||η||Lp

k

k+1

≤z6||α||Lp||η||Lp.

k+1

k+1

(3.40)

Thus,ifαissmallenough,η=0.

ˆ(E).SupposethatH˜2Corollary3.8.Let(A,u)∈D(A,u)=0.Thenevery

1˜u(A,˜)∈H(A,u)(Y,End(E))

ˆ(E)of(A,u).comesfroma1-parameterfamilyofvariations(At,ut)∈D

10,31proof.SinceH(α,υ˜)∈A,u)(Y,End(E))≃H⊕H,weprovethatforany(˜

H1⊕H0,3thereexists(αt,υt)∈Ω1⊕Ω0,3withα0=0,υ0=0suchthat

+

αt+P+(αt∧αt)+B(u,υt)=0,DA

∗¯ADυt+Λ(αt)(u+υt)=0,

andH(α˙)=α˜,H(υ˙)=υ˜,whereα˙:=ddtυt|t=0.

Iftissmallenough,thenthereexistsaunique(αt,υt)∈S(A,u)suchthat

∗

tα˜=αt+D+◦G(2)◦(P+(αt∧αt)+B(u,υt)),

′¯Atυ˜=υt+D◦G(2)(Λ(αt)(u+υt)).

Clearly,α0=υ0=0,andtα˜=H(αt),tυ˜=H(υt).Differentiatingthese

withrespecttot,weobtain

α˜=H(α˙),υ˜=H(υ˙).

˜0H(At,ut)=0followsfromtheuppersemi-continuityofthecohomology

oftheellipticcomplex.

References

[AHS]

M.F.Atiyah,N.J.Hitchin,andI.M.Singer.Self-dualityinfour-dimensionalRiemanniangeometry.Proc.Roy.Soc.LondonSer.A,362(1711):425–461,1978.

LaurentBaulieu,HiroakiKanno,andI.M.Singer.Specialquan-tumfieldtheoriesineightandotherdimensions.Comm.Math.Phys.,194(1):149–175,1998.

17

[BKS]

[DGNV]

RobbertDijkgraaf,SergeiGukov,AndrewNeitzke,andCumrunVafa.TopologicalM-theoryasunificationofformtheoriesofgravity.Adv.Theor.Math.Phys.,9:593–602,2005.

S.K.DonaldsonandR.P.Thomas.Gaugetheoryinhigherdimensions.InThegeometricuniverse(Oxford,1996),pages31–47.OxfordUniv.Press,Oxford,1998.

DanielS.FreedandKarenK.Uhlenbeck.Instantonsandfour-manifolds,volume1ofMathematicalSciencesResearchInstitutePublications.Springer-Verlag,NewYork,secondedition,1991.A.Iqbal,N.Nekrasov,A.Okounkov,andC.Vafa.Quantumfoamandtopologicalstrings.preprint,hep-th/0312022,2003.Hong-JongKim.ModuliofHermite-Einsteinvectorbundles.Math.Z.,195(1):143–150,1987.

ShoshichiKobayashi.Differentialgeometryofcomplexvectorbundles,volume15ofPublicationsoftheMathematicalSocietyofJapan.PrincetonUniversityPress,Princeton,NJ,1987.KanoMemorialLectures,5.

JunLiandGangTian.VirtualmodulicyclesandGromov-Witteninvariantsofalgebraicvarieties.J.Amer.Math.Soc.,11(1):119–174,1998.

NikitaNekrasov.Z-theory:chasingm/ftheory.C.R.Phys.,6(2):261–269,2005.Strings04.PartII.

NikitaNekrasov,HirosiOoguri,andCumrunVafa.S-dualityandtopologicalstrings.J.HighEnergyPhys.,(10):009,11pp.(electronic),2004.

CliffordHenryTaubes.Casson’sinvariantandgaugetheory.J.DifferentialGeom.,31(2):547–599,1990.

R.P.Thomas.AholomorphicCassoninvariantforCalabi-Yau3-folds,andbundlesonK3fibrations.J.DifferentialGeom.,54(2):367–438,2000.

GangTian.Gaugetheoryandcalibratedgeometry.I.Ann.ofMath.(2),151(1):193–268,2000.

[DT]

[FU]

[INOV][Ki][Ko]

[LT]

[N][NOV]

[Ta][Th]

[Ti]

18

DepartmentofMathematics,M.I.T.

77MassachusettsAvenue,CambridgeMA02139,USA.e-mail:tanaka@math.mit.edu

19