arX

iv:h

ep-t

h/99

1208

6v3

30

Dec

199

9

hep-th/9912086

Duality Relations

Am ong TopologicalE�ectsIn String Theory

Edward W itten �

PhysicsDepartm ent,California Institute ofTechnology,Pasadena CA 91125 USA

and

CIT-USC Center forTheoreticalPhysics,Univ.ofSouthern California,LosAngelesCA

W e explore two di�erent problem s in string theory in which duality relates an ordinary

p-form �eld in one theory to a self-dual(p+ 1)-form �eld in another theory. One prob-

lem involves com paring D 4-branes to M 5-branes,and the other involves com paring the

Ram ond-Ram ond form s in Type IIA and Type IIB superstring theory. In each case,a

subtle topologicale�ect involving the p-form can be recovered from a carefulanalysisof

the quantum m echanicsofthe self-dual(p+ 1)-form .

� On leavefrom InstituteforAdvanced Study,Princeton NJ 08540

Decem ber,1999

1. Introduction

Thepurposeofthispaperistoexplorehow certain relatively subtletopologicale�ects

in string theory and M -theory transform into each otherunderdualities.W e willlook at

two casesthatare rathersim ilarand can betreated in rough parallel:

(1)The\U (1)gauge�eld"on theworld-volum eofaTypeIID -braneisactuallybetter

described asa Spincstructure (assum ing,aswe generally willin the presentpaper,that

the background Neveu-Schwarz three-form �eld H is topologically trivial). This e�ect,

which �rstshowed up in a detailed exam ple [1],hasa naturalinterpretation in K -theory

[2,3]and can bedem onstrated by studying globalanom aliesforelem entary stringsending

on the D -brane [4]. The e�ect existsforType IIA and IIB D p-branes forseveralvalues

ofp.The problem we willstudy arisesin the case ofa Type IIA D 4-brane.Such a brane

can arise upon com pactifying an M 5-brane on a circle,in which case the \gauge �eld" of

theD 4-branearisesby com pactifying thechiraltwo-form (with self-dualcurvature)on the

M 5-brane.Itm ustsom ehow bepossibleto deduce theSpincnatureoftheD 4 gauge�eld

from som eproperty ofthe chiraltwo-form ofthe M 5-brane.

(2)TheRam ond-Ram ond four-form �eld strength G 4 ofTypeIIA superstring theory

doesnot,in general,obey conventionalDirac quantization. Undercertain conditions[5],

thereisa gravitationalcorrection to thequantization law,and theperiodsofG 4 arehalf-

integral.TypeIIA superstringtheory on aspacetim eX = S1� Y isT-dualtoTypeIIB on

thesam espacetim e.TheT-duality m apstherelevantpartofG 4 to theself-dual�ve-form

G 5 ofType IIB on S1 � Y . Hence,in this situation,it m ust be possible to deduce the

nonintegrality ofthe G 4 periodsfrom som eproperty ofthe dynam icsofG 5.

W hattheseexam pleshavein com m on isthaton onesideoftherelation,oneconsiders

a �eld (the \gauge �eld strength" on the D 4-brane,orthe four-form ofType IIA)whose

periodsareshifted from conventionalDiracquantization by a gravitationalcorrection.On

the otherside ofthe relation isa self-dualBose�eld ofone degree higher(the three-form

ofthe M 5-brane,and the �ve-form ofType IIB)in a related theory. W e m ustsom ehow

deduce the gravitationalcorrection in the lowerdim ension from the quantum m echanics

ofthe self-dual�eld in the higherdim ension.

Thequantum m echanicsofa self-dual�eld isquitesubtleand hasbeen studied from

m any pointsofview,a sam pling being [6-29]. Recent work has included construction of

braneLagrangiansatleastlocally[20,24,25]and construction ofm anifestlysupersym m etric

and kappa-sym m etricequationsofm otion form ultipletsincluding theself-dual�elds[26].

1

As is m ost fam iliar from the case ofa chiralscalar (self-dualone-form ) in two di-

m ensions,and aswewillreview in section 3,a chiralp-form �eld generally hason a given

m anifold severalpossiblepartition functions,determ ined by achoiceoftheta function;one

needsa recipeto pick outtherighttheta function in a given situation.Forp > 1,thishas

been dem onstrated m ost explicitly in [30]. The right recipe for picking a theta function

dependson som ephysicalinput;fortheself-dualthree-form oftheM 5-brane,and theself-

dual�ve-form ofTypeIIB,a prescription hasbeen given in [31].Foronespeci�cexam ple

abovetwo dim ensions{theself-dualthree-form on T 6,wherethe partition function turns

out to be unique (independent ofthe spin structure on T 6) { the appropriate partition

function has been constructed and studied in detail[32]. The recipe of[31]for picking

a theta function has been related to a m ore classicaltopologicalinvariant (the Kervaire

invariant)in [33].

An exception to the statem ent that the chiralp-form has severalpossible partition

functionsarises[8]ifonecom binesseveralchiralbosonsusing an even unim odularlattice.

Then one getscom plete m odularinvariance and a unique partition function.Thiscase is

very im portantfortheheteroticstring [34].In a di�erentcase(likea singlechiralscalarat

thefreeferm ion radius,relevantto thepresentpaper),onecannotresolvetheam biguity of

the partition function by sum m ing overallpossibilitiesbecause each candidate partition

function hasslightlydi�erentanom alies,and itdoesnotm akesensetoadd them .In theM -

theory and TypeIIB applications,thechiralp-form doesnotappearby itselfbuttogether

with addition �elds such as ferm ions. The com plete partition function is presum ably

anom aly-free(thishasnotbeen com pletely dem onstrated);anom aly cancellation depends

on pairing the proper (spin-structure dependent) partition function ofthe ferm ionswith

the properpartition function ofthe chiralp-form .Thus,one m ustexpectthatthe recipe

for picking a chiralp-form partition function depends on the spin structure,and this is

the case forthe proposalin [31]. Once the anom aliesare allcanceled,itispossible,and

perhapscorrectphysically,to sum overspin structures.

Them ain goalofthepresentpaperistoshow how thequantum m echanicsoftheself-

dual�eldsgivesrise,aftercom pacti�cation on a circle,to thee�ectsm entioned in (1)and

(2)above.In section 2,wedem onstrate the phenom ena in specialcasesin which detailed

generaltheory isnotneeded.In therestofthepaper,weproceed m oresystem atically.In

section 3,we recallsom e im portantfactsaboutp-form quantum m echanics.In sections4

and 5,wem akethetheory in [31]m oreconcreteforthesituation ofinterestand useitto

deduce whatwe need.

2

The�rstofourtwoproblem sdescribed aboveissom ewhatrem iniscentoftheproblem

ofrelating them echanism ofM 5-branenorm albundle anom aly cancellation [35]with the

corresponding m echanism in TypeIIA [31].Therelation between them hasbeen analyzed

recently [36].

2. R eduction To C hiralScalar

The goalin the present section is to verify that the phenom ena m entioned in the

introduction work out correctly in som e sim ple cases in which we can do this without

m any technicalities.Thiswillperhapssatisfy the curiosity ofsom e readers,and m ay give

othersthecourageneeded to perseverethrough thetechnicalitiesoftherestofthepaper.

M 5-Brane W rapped On A Circle

W e�rstconsidertherelation oftheM 5-branetotheD 4-brane.Ourgoalistoanalyze

the M 5-brane on a world-volum e V = S � R,where R isan oriented �ve-m anifold and S

isa circle with a supersym m etry-preserving spin structure. To do thisin generalwillbe

thegoalofsection 5,butthingsarem uch sim plerin thecaseR = eS � R 0,with eS another

circleand R 0 a four-m anifold.Thesim plicity willarisebecause in thisspecialcase,wedo

notneed to understand chiralp-form s�eldsofp > 0;we can deduce whatwe need from

fam iliar(though subtle)factsaboutchiralscalars.

Though wecould treatan arbitraryR 0,itwillsu�ceforillustration totakeR 0= C P2.

Thus,the �vebrane world-volum e willbe V = �� C P2where � = S � eS isa productof

circles;the spin structure on S preserves supersym m etry buteitherchoice m ay be m ade

on eS.The nontrivialcohom ology group ofC P2(apartfrom dim ensionszero and four)is

H 2(C P2;Z)= Z: (2:1)

The generatorofH 2(C P2;Z)isa self-dualform ! thatobeys

Z

C P1

! = 1;

Z

C P2

! ^ ! = 1: (2:2)

Here C P1isa linearly em bedded subspace ofC P

2and generatesH 2(C P

2;Z).

W esupposethattheM -theory spacetim eisX = �� C ,whereC isanine-dim ensional

spin-m anifold in which C P2is em bedded. M -theory on this spacetim e is equivalent to

Type IIA on X 0 = eS � C ;the M 5-brane corresponds to a D4-brane wrapped on R =

3

eS � C P2. R isnota spin m anifold,since C P

2isnot. Asa result,according to [4],the

�eld strength F ofthe \U (1) gauge �eld" on the D4-brane does not obey conventional

Diracquantization.Rather,Z

C P1

F

2�= n +

1

2; (2:3)

with integern.

Thegauge�eld on theD4-branearisesby dim ensionalreduction from thechiraltwo-

form b on the M 5-brane. W e want to know how (2.3)arises from the theory ofa chiral

two-form . W e consider a lim itin which the radiiofS and eS are m uch greater than the

size ofthe C P2. In this case,the physics on the M 5-brane reduces to an e�ective two-

dim ensionaltheory on � = S � eS. In fact,the �eld b reduces (by the ansatz b = !�)

to a chiralscalar� in two dim ensions.� appearsatthe self-dualorfree ferm ion radius1;

the � �eld ishence equivalentquantum m echanically to a com plex ferm ion ofpositive

chirality.

The �eld propagates on the Riem ann surface �,and the partition function of

dependson a choice ofspin structure on �. So to describe the physics,we need to know

the e�ective spin structure on � in the low energy theory,given the underlying choice of

spin structureon theM -theory spacetim eX = �� C .Sincechoosing a spin structureon

X isequivalenttochoosingaspin structureon �and choosingoneon C ,in them icroscopic

M -theory description a spin structure was chosen on � atthe beginning. In fact,as we

noted above,we are interested in the case that this spin structure is the product ofthe

supersym m etric spin structure on S and any desired spin structure on eS. It is natural

to guess that the e�ective spin structure on � in the low energy theory is just the spin

structureon � thatwestartwith m icroscopically.Thisassertion alm ostfollowsjustfrom

the fact that the m ap from the m icroscopic to the m acroscopic spin structure m ust be

1 In general,ifthe M 5-brane is com pacti�ed to two dim ensions on a four-m anifold R0,the

chiraltwo-form reduces to a set oftwo-dim ensionalscalars with m om entum lattice given by the

two-dim ensionalcohom ology lattice ofR0. For R

0= T

4,this assertion is built into the detailed

com putation in [32].ForR0= C P

2,the lattice isone-dim ensional,generated by a vector! with

!2= 1;thisisthe lattice ofa chiralboson with the free-ferm ion radius. (D epending on how R

0

isem bedded in the fullspacetim e,som e ofthe conservation lawsassociated with the m om entum

lattice m ay be violated by instantons constructed from m em branes with boundary on R0. This

phenom enon is irrelevant for determ ining the �vebrane partition function in the large volum e

lim it.)

4

invariantunder the action ofSL(2;Z)on �,and can be veri�ed using the techniques of

sections4 and 5.

In thetheory ofa D 4-braneon eS � C P2,with eS regarded asthe\tim e" direction,the

ux (2.3)can beinterpreted asa conserved charge.G oing back to theself-dualthree-form

theory on the M 5-brane worldvolum e V = S � eS � C P2,this ux is interpreted as the

integralofthe self-dualthree-form T (which is the curvature ofthe chiraltwo-form b,

de�ned by T = db)overS � C P1. In term softhe ansatz b= !�,we have T = ! ^ d�,

and theconserved charge is

q=

Z

S�C P2

! ^ d�

2�=

I

S

d�

2�: (2:4)

In thefreeferm ion description,d�=2� becom es and thechargeistheconserved ferm ion

num ber

q=

I

S

: (2:5)

Now,since the ferm ions on S are in the supersym m etric spin structure,both and

have a single zero m ode on S. The quantization ofthe zero m odes gives rise,in a way

thatisfam iliarfrom the Ram ond sectorofsuperstrings,to a two-fold degeneracy ofthe

ground state. The ground states have ferm ion num ber q = �1=2,and allexcited states

havehalf-integraleigenvaluesofq.Sinceqisinterpreted in theTypeIIA description asthe

ux in (2.3),wehaveexplained thehalf-integrality ofthat ux starting with thetheory of

the self-dualthree-form on theM 5-brane.

It is also instructive to consider,in a sim ilarfashion,a case in which the D 4-brane

iswrapped on a �ve-m anifold R thatdoesnothave a Spincstructure,so thatthe theory

should beinconsistent.Such acaseisobtained bytakingR tobenotaproducteS� C P2but

a C P2bundleover eS in which the�berundergoescom plex conjugation in going around eS.

Com plex conjugation reversesthesign of! and so actson � by � ! ��.Theperiodsof�

thusm ustchangesign in goingaround eS,butsincetheyarehalf-integral,thisisim possible.

Thisistheinconsistency.Butwhatdoesitlook likein thefreeferm ion description? From

thispointofview,� ! �� is $ .Alternatively,if = ( 1 + i 2)=p2 with M ajorana-

W eylferm ions 1, 2,itis

1 ! 1; 2 ! � 2: (2:6)

Both 1 and 2 couple to the supersym m etric spin structure on S,and in view of(2.6),

they see opposite spin structures on eS. So 1 and 2 together have precisely one zero

5

m ode on S � eS. Having an odd num berofferm ion zero m odesm eansthatthe partition

function vanishes,and thatthisvanishing cannotbelifted by insertionsoflocaloperators

(a ferm ionic operator willnot have an expectation value once we average over spatial

rotations).Itshould be interpreted asa kind ofglobalanom aly.W e willargue in section

5.1 thatthe M 5-branehassuch an inconsistency on S � R wheneverR isnotSpinc.

Analog For Type IIB

Now let us brie y discuss the analogous issues in the other case m entioned in the

introduction.

Our goalis to com pare topologicale�ects in Type IIB and Type IIA superstring

theory on S � Y ,with S a circleand Y a nine-dim ensionalspin m anifold.Buta shortcut

along the above lines is possible for the specialcase Y = eS � Y 0,with eS another circle

and Y 0 an eight-dim ensionalspin m anifold. So we consider Type IIB superstring theory

on S � eS � Y 0,with the supersym m etric spin structure on the�rstfactor.

Forillustration,weconsiderthecasethatY 0= H P2.Theonly nontrivialcohom ology

group ofthism anifold isH 4(H P2;Z)= Z.The generatorisa self-dualfour-form ! such

that Z

H P1

! = 1;

Z

H P2

! ^ ! = 1: (2:7)

Here H P1isa linearly em bedded subspace ofH P

2and generatesH 4(H P

2;Z).H P

2isa

spin m anifold,so its�rstPontryagin classp1 isdivisibleby 2,and � = p1=2 obeys

Z

H P1

� = 1: (2:8)

In fact,� isjust!.

W ecan repeatm uch ofwhatwehavealreadyseen.In com pacti�cation on S� eS� H P2,

with the lastfactorm uch sm allerthan the �rsttwo,the chiralfour-form C 4 ofType IIB

superstring theory reduces at long distances (via an ansatz C4 = !�) to a chiralscalar

� on S � eS. � can be expressed in term soffree ferm ions,and by the sam e reasoning as

above,ifwe regard eS asthe\tim e" direction,then the conserved charge

q=

I

S

d�

2�(2:9)

takeshalf-integralvalues.One can think ofq m ore m icroscopically as

q=

I

S�H P1

G 5

2�(2:10)

6

where G 5 = dC4 isthe gauge-invariantself-dual�ve-form ofTypeIIB.

Now we consider a T-duality transform ation on the �rst circle S. This m aps Type

IIB superstring theory to Type IIA,and the m odes ofG 5 thatappear in the integralin

(2.10)are m apped to G 4,the Ram ond-Ram ond four-form �eld strength ofType IIA.In

the TypeIIA description,q becom es

q=

Z

H P1

G 4

2�: (2:11)

Thus,to account for the half-integrality ofq from the Type IIA point ofview,we m ust

explain why G 4 hashalf-integralperiodsin thissituation.

Butthisisa consequence of(2.8).The generalform ula isindeed [5]

Z

U

G 4

2�=1

2

Z

U

� + integer; (2:12)

forany four-cycle U in a Type IIA spacetim e. In view of(2.8),thisisequivalentto half-

integrality ofq.Thus,wehavesucceeded,in thissituation,in reconciling thegravitational

shiftin thequantization law ofthefour-form in TypeIIA with thesubtletiesoftheself-dual

�ve-form ofTypeIIB.

Fora m orecom pletestudy oftheseproblem s,wherewecom pactify on only onecircle

and nottwo,we need to delve into the theory ofchiralp-form �eldsforp > 0. Thiswill

bethesubjectofsections4 and 5.But�rstwem ustrecallsom eadditionalaspectsofthe

quantum m echanicsofself-dualp-form s,starting with theone-form case.

3. Q uantum M echanics O fSelf-D ualp-Form s

Before looking at our speci�c problem , we need som e m ore background on chiral

p-form s.

In constructingthequantum m echanicsofan ordinary (notself-dual)p-form �eld on a

m anifold M ,onesum soverallperiodsin H p(M ;Z).Thatisnotso fora self-dualp-form .

In fact,itisim possibleto im poseany classicalquantization law atallon the periods

ofa self-dualp-form . To illustrate this,let� be a two-torusconstructed asC =�,where

C isthecom plex z-plane,and � isa latticegenerated by com plex num bers1 and � (with

Im � > 0).LetA bea cyclein � thatliftsin C to a path from 0 to 1,and letB bea cycle

thatliftsto a path from 0 to �. Let� be a self-dualone-form . Then � = cdz for som e

7

com plex constant c. Ifwe want,for exam ple,R

A�=2� to be integral,we need c 2 2�Z,

whilerequiringR

B�=2� to be integralputsan entirely di�erentcondition on c.

W hat happens instead is thata self-dualp-form m ust be treated quantum m echan-

ically;one cannot treat its periods classically. The partition function ofsuch a �eld is

written asa sum overonly halfthe periods. Forillustration,letusconsideran exam ple

[8]thatisextrem ely im portantin string theory:a collection of8k chiralbosons�i in two-

dim ensions,forsom eintegerk,associated with an even unim odularlattice� with positive

de�niteintersection form (;).W eset�i = d�i.Thepartition function in genusoneisas

follows. Let� be asabove and q = exp(2�i�). Then the partition function ofthe chiral

boson theory on � is

Z(q)=

P

w 2�q(w ;w )=2

�(q)8k(3:1)

with � theDedekind eta-function.In thisform ula,thepartition function isconstructed as

a sum overa singlesetofperiods{ theperiodswi =R

A�i=2�,which arethecom ponents

ofasinglelatticevectorw 2 �.In aHam iltonian fram ework with A regarded asthespatial

cycle and B astim e,the A-periodslabelthe winding (orby self-duality the m om entum )

states;thetheta function in thenum eratorof(3.1)com esfrom thesum overthesestates.

Ofcourse,the choice ofthe particularcycle A isnotuniquely determ ined. The partition

function isSL(2;Z)-invariant;by an SL(2;Z)transform ation,onecould replacethecycle

A by nA + m B forany relatively prim e integersn;m .

Intuitively,we m ay think oftwo periodsR

A� and

R

A 0 � as com m uting ifand only

ifthe intersection num ber A \ A 0 is zero. There is no way to sim ultaneously m easure

noncom m uting periods.Thepartition function isconstructed asa sum overa m axim alset

ofcom m uting periods.

The exam ple relevantto the presentpaperisslightly m ore subtle:itisthe case that

thechiralbosons�i arederived from a lattice� thatisunim odular,butnoteven.In fact,

theprototypeforusisa singlechiralboson atthefreeferm ion radius,thatisto say � isa

one-dim ensionallatticegenerated by a vector! with (!;!)= 1.In thiscase,there isnot

a single partition function;rather (asis apparent from the description by free ferm ions)

there isa partition function foreach choice ofspin structure.Itisinstructive to exam ine

thesepartition functions.They areconveniently written in term sofstandard functionsas

Z

��

�

�

(zj�)=

#

��

�

�

(zj�)

�(�); (3:2)

8

where� and � are0 and 1/2 and z isan extra variableincluded to representthecoupling

to a background gauge�eld.Thepartition function in theabsenceofthis�eld isobtained

by setting z = 0. The functionsin the num eratoron the righthand side are called theta

functionswith characteristics.They areexplicitly

#

�0

0

�

(zj�)=X

n2Z

qn2=2 exp(2�inz)

#

�0

1=2

�

(zj�)=X

n2Z

(�1)nqn2=2 exp(2�inz)

#

�1=2

0

�

(zj�)=X

n2Z + 1=2

qn2=2 exp(2�inz)

#

�1=2

1=2

�

(zj�)= iX

n2Z + 1=2

(�1)n+ 1=2qn2=2 exp(2�inz):

(3:3)

W e have written these theta functions as sum s over the A-period n =R

Ad�=2�. By

SL(2;Z),one could instead write each ofthese theta functions as a sum over any other

chosen period ofd�. W hile #

�1=2

1=2

�

, which corresponds to the odd spin structure, is

SL(2;Z)-invariant (up a a c-num ber m ultiple that re ects the m odular weight plus an

anom alousphase),theothersareperm uted by SL(2;Z),so ifonechoosesto write#

�0

0

�

,

forexam ple,with a di�erentchoice ofthe period,one m ighthave to use the form ula for

#

�1=2

0

�

.

In constructing the theta function as a sum over the values ofthe A-period n,this

period is integralfor � = 0 and half-integralfor � = 1=2. Therefore the answer to the

question ofwhether a given period ofthe self-dualone-form is integralor half-integral

dependson thechoiceoftheta function.On theotherhand,� determ inesthesign factors

in the sum over the A-periods. A con�guration with a given value ofthe A-period n is

weighted by a sign +1 if� = 0 and by a sign (�1)n (or(�1)n+ 1=2 ifn ishalf-integral)if

� = 1=2.

Now,wewanttodescribethethetafunctionsin away thatgeneralizestohighergenus

surfacesand also to self-dualp-form sofp > 1. W e willde�ne a Z 2-valued function (x)

on the lattice� asfollows.2 Forthe latticepoints1 and �,we set

(1)= (�1)2�; (�)= (�1)2�: (3:4)

2 In [31],thisfunction wascalled H (x),butIwantto avoid notationalclasheswith the three-

form �eld H ofstring theory and Hi(M )forcohom ology groups.

9

W e extend to a function on thewhole latticeby requiring

(x + y)= (x)(y)(�1) (x;y); (3:5)

where(x;y)= �(y;x)istheintersection form on thelattice�.Forexam ple,thisde�nition

gives

(1+ �)= �(1)(�); (3:6)

since 1 and � correspond to the cycles A and B ,whose intersection num ber is 1. (3.5)

isthe basic form ula. Theta functions are in naturalone-to-one correspondence with Z2-

valued functionson thelatticethatobey thisrelation.G iven ,thecharacteristics�;� are

extracted from (3.4)and used to write the explicitform ulasforthe theta functions that

we gaveabove.

Let�1 and �2 be,respectively,the sublatticesof� generated by 1 and by �;we call

these the A-lattice and the B -lattice. As we saw above,a con�guration with A-period

n contributesto the theta function (in the representation ofthatfunction asa sum over

the A-periods) with a sign 1 or (�1)n depending on �. (3.4) m eans that (x) for x in

the A-lattice issim ply the sign factor with which a con�guration ofA-period n = x (or

n = x+ 1=2)contributesto thetheta function.Likewise,wesaw abovethat� determ ines

whether the A-periods are integralor half-integral,and thus this is determ ined by (x)

forx in theB -lattice.

The classi�cation oftheta functionsby Z 2-valued functions(x)extendsbeyond the

genusonecasethatwehavejustconsidered:levelonetheta functionsofany lattice� with

unim odularantisym m etric form (; )and a m etric forwhich thisform ispositive and of

type(1;1)areclassi�ed by functionsobeying(3.5).Thisfacthasadi�erential-geom etric

explanation thatwasreviewed in [31].(Thebasicidea isthatsuch an determ inesa line

bundleover�;thislinebundlehasup to constantm ultiplesa uniqueholom orphicsection

which is the theta function.) For our present purposes, we willsim ply note that the

functions thatobey (3.5)transform underSL(2;Z)the sam e way thattheta functions

do.In thisassertion,thesign factor(�1)(x;y) in (3.5)isessential.Forexam ple,thetheta

function #

�1=2

1=2

�

associated with the odd spin structure isSL(2;Z)-invariant,so itm ust

beassociated with a function (x)thatislikewiseSL(2;Z)-invariant.Since� = � = 1=2,

thistheta function has(1)= (�)= �1.AsSL(2;Z)can m ap the latticepoints1 or�

to 1+ �,itfollowsthat(1+ �)m ustequal�1,which iswhatwe getfrom (3.6).

10

To write the fourtheta functionsby explicitform ulasasin (3.3)requiresa choice of

A-lattice. Som e m ore inform ation is needed,though,because the choice ofA-lattice is

invariantunder� ! � + 1,butthisoperation perm utesthetheta functionsin a non-trivial

fashion. Ifone isalso given a choice ofB -lattice (and thusessentially the basis(1;�)for

the lattice �),thisism ore than enough inform ation to enable the writing ofthe explicit

form ulasin (3.3).(Forthat,itisenough to know the B -cyclesm od 2.) Ifone haschosen

both the A-lattice and the B -lattice,then one has an explicit SL(2;Z) transform ation

� ! �1=� thatexchangesthem .Itexchanges� and �,and thusexchangesa half-integral

shiftin thevalueofthe A-period n with a sign factorby which the di�erentvaluesofthe

A-period areweighted.

Generalization

Now let us consider the generalization to a self-dualp-form �eld G p,ofp possibly

biggerthan 1,on a2p-dim ensionalm anifold M .(Foradetailed treatm entviaholom orphic

factorization ofthe partition function ofa non-chiraltheory,see [30].) The periodstake

values in � = H p(M ;Z),which for sim plicity we willassum e to be torsion-free. Thus

� is a lattice,with an antisym m etric bilinear form ( ; ) ofdeterm inant 1 that is given

by the intersection pairing on M . If� has rank 2g,then it has has 22g distinguished

theta functions#

��

�

�

(zj�)thatwewillintroducem om entarily.Thepartition function of

G p is #

��

�

�

(zj�)=�,where � (analogous to �(�) in (3.2))is uniquely determ ined from

the non-zero m odes ofG . The subtlety com es from the choice oftheta function in the

num erator.

Asin thecaseofaone-form �eld,theperiodsofG arenotallsim ultaneously m easure-

able.The bestthatone can do isto pick a m axim alsublattice �1 consisting ofm utually

\com m uting" periods. �1 is a lattice ofA-periods,that is,it is a half-dim ensionalsub-

lattice of� such that(x;y)= 0 forx;y 2 � 1. Itisconvenient,though notnecessary,to

pick also a com plem entary lattice�2 ofB -periods.Thus,� = � 1 � �2,and (x;y)= 0 for

x;y 2 �2. Picking the B -periodsand A-periodsgivesan explicitperiod m atrix �ij = �ji,

i;j= 1;:::;g forthe lattice�.

Once the A-cycles and B -cycles are �xed,one can write an explicitform ula for the

theta functions. One picks a half-lattice vector � 2 1

2�1=�1,and a half-lattice vector

� 2 1

2�2=�2.The theta function with characteristics�,� isthen

#

��

�

�

(zij�)=X

n2� 1+ �

exp

0

@ i�X

ij

ninj�ij + 2�ini(zi+ �i)

1

A : (3:7)

11

The zi are param eters that m easure the coupling to a background p-form potential;the

partition function isobtained by setting zi = 0 (and dividing by �).

From (3.7),we see thatifwe write the theta function asa sum overA-periods,then

the A-periods are shifted from integers by � 2 1

2�1=�1. But the sign factor in the sum

overA-periodsisdeterm ined by �.

As in the g = 1 case thatwe discussed �rst,the theta functions are m ostnaturally

classi�ed by aZ 2-valued function (x)on thelattice�thatobeysthefundam entalrelation

(x + y)= (x)(y)(�1) (x;y) forallx;y 2 �: (3:8)

G iven such a function,one de�nesthecharacteristics�;� by

(x)= (�1)2(x;�) ifx 2 �1

(x)= (�1)2(x;�)ifx 2 �2;(3:9)

and then the theta function can be de�ned by the form ula in (3.7).Asm entioned above,

there is also a m ore intrinsic procedure to go from to the theta function (use to

constructa linebundle and takeitsholom orphic section).

Com bining the above de�nitions,we can see how the theta function depends on .

Forx 2 �1,(x)isa sign factorin thesum overA-periods,and forx 2 � 2,(x)controls

the non-integrality oftheA-periods.Thisgeneralizeswhatwe explained above forg = 1.

Specializing to M = S � F

In general, this form alism is som ewhat abstract, partly because for a general2p-

dim ensionalm anifold M ,there isno particularly nice choice ofA-periodsand B -periods.

Nicechoicesdo existifM = S � F ,with S a circleand F a m anifold ofdim ension 2p� 1.

Thiscase isourfocusin the presentpaper. The theory ofa self-dualp-form G p on such

an M reduces at low energies on F to a theory ofan ordinary p-form G 0

p with no self-

duality,or(aftera duality transform ation)to a theory ofan ordinary (p� 1)-form G 0

p�1 .

Correspondingly,thecohom ology ofM splitsas

H p(M ;Z)=H p(F ;Z)� H 1(S;Z) Z Hp�1 (F ;Z)

=H p(F ;Z)� H p�1 (F ;Z):(3:10)

W etake�= H p(M ;Z),so thata partition function ofG p on M isa theta function of�,

and weset�1 = H p(F ;Z),�2 = H p�1 (F ;Z).

12

W e take the lattice ofA-periodsto be �1 and the lattice ofB -periodsto be �2. A

theta function for � can be constructed either as a sum over A-periods { corresponding

to the representation ofthe theory on F in term sofG 0

p { oras a sum overB -periods{

corresponding to the representation ofthetheory on F in term sofG 0

p�1 .

No m atter which representation one uses,the theta function of� is determ ined by

a choice of a suitable function (x) on �. How to construct such a function for the

self-dualp-form �elds m entioned in the introduction was explained in [31]. Once is

selected,itsrestriction to �1 determ inesa sign factorin the sum overperiodsifone uses

thedescription ofthetheory in term sofG 0

p,ora nonintegrality oftheG0

p�1 periodsin the

otherdescription. Conversely,the restriction of to � 2 determ inesthe nonintegrality of

G 0

p periods,ora sign factorin thesum overG 0

p�1 .

The factthata sign factoron oneside becom es,afterduality,a nonintegrality ofthe

periods in the other description can also be explained m ore m icroscopically in term s of

a Feynm an path integralrepresentation ofthe G 0

p and G 0

p�1 theories. M ore generally,in

d dim ensions,a phase factor in a theory with k-form curvature G k is always translated

afterduality to a shiftin periodsofa dual�eld G d�k .Thisholdsforalld and k.A path

integralderivation ofthisfactcan be found (ford = 4,k = 2,butthe generalcase isnot

essentially di�erent)in section 4.2 of[37].

4. System atic A nalysis O fType II C ase

In attem pting a system atic treatm ent,using the fram ework of[31],ofthe problem s

m entioned in theintroduction,wewillbegin with thesecond problem { understanding the

shifted quantization law ofthe Type IIA four-form from the quantum m echanics ofthe

self-dual�ve-form ofTypeIIB.Thiscase involvesfewertechnicalities.

4.1.Outline

In TypeIIB theory on a ten-dim ensionalspin m anifold X ,wehavea four-form poten-

tialC4 with a self-dualcurvature�ve-form G 5.Ifwecould om ittheself-duality condition,

and we im pose conventionalDirac quantization,then the C4-�eldsare classi�ed topologi-

cally by a classx 2 H 5(X ;Z).Here x isrepresented in de Rham cohom ology by G 5=2�.

W e som etim eswriteinform ally x = [G 5=2�].

Foran ordinary four-form �eld,wewould constructthepartition function by sum m ing

overallchoicesofx (and foreach choiceofx,integratingoverallpossibilitiesforC4).Fora

13

four-form ofself-dualcurvature,wedo notsum independently overallvaluesofx.Rather,

asdiscussed in section 3 aboveand in [31],weconstructthepartition function in term sof

a theta function on T = H 5(X ;U (1)),which,ifthere isno torsion in the cohom ology of

X ,isthe torusH 5(X ;R )=H 5(X ;Z).The theta function,aswe discussed in section 3,is

constructed by sum m ing overa m axim alsetof\com m uting" periods.3

To construct the theta function, as explained in [31]and in section 3, we need a

function (x)from H 5(X ;Z)to the group Z2 = f�1g,obeying

(x + y)= (x)(y)(�1) x�y (4:1)

forallx;y 2 H 5(X ;Z).Here x � y isthe intersection pairingR

Xx [ y. The function is

needed to determ ine a linebundle on T ,a suitable section ofwhich isthe theta function.

Itisconvenientto write(x)= (�1)h(x),whereh(x)isan integer-valued function thatis

de�ned m odulo two.

In whatfollows,wewillstudy thefunction h(x)forthecasethatX = S � Y ,with S a

circleendowed with a spin structureofunbroken supersym m etry (thatis,a non-bounding

spin structure)and Y an arbitrary nine-dim ensionalspin-m anifold.W e will�nd thatifx

isan elem entofH 5(Y ;Z),then

h(x)=

Z

Y

� [ x; (4:2)

where � isthe integralcharacteristic classsuch that2� = p1(Y ). H5(X ;Z)isgenerated

by H 5(Y ;Z)togetherwith elem entsoftheform a[ w with a a generatorofH 1(S;Z)= Z

and w 2 H 4(Y ;Z).So the function (x)would be com pletely determ ined by (4.2),(4.1),

and a knowledge ofh(a [ w). It does not seem that there is a form ula for h(a [ w) as

elem entary as(4.2).4 Itturnsout,though,that(4.2)su�cesto answerthequestion raised

in the introduction.

3 In general,H5(X ;U (1))hascom ponentslabeled by the torsion subgroup ofH

6(X ;Z);each

com ponent is a torus. This re�nem ent willnot be essential in our present discussion, and I

suspect that the torsion can be fully taken into account only ifone workswith K -theory rather

than cohom ology,a task that we initiate in section 4.3 below. Note that in what follows,we

writetheproductofdi�erentialform sasa wedgeproduct,denoted ^,and theproductofintegral

cohom ology classesasa cup product,denoted [.

4 For exam ple,(4.2) shows that h(x) is independent ofthe spin structure for x 2 H5(Y ;Z);

but exam ples such as Y = eS � Y0(with eS another circle and Y

0an eight-m anifold) show that

h(a[ w)doesdepend on the spin structure.In thiscase,forw 2 H4(Y

0;Z),h(a[ w)isequalto

R

Y 0� [ w or 0 depending on whether one takes the supersym m etric or nonsupersym m etric spin

structure on eS.Thiscan be seen by the m ethodsofsection 2.

14

Thiscom esaboutasfollows.Theuseofthefunction (x)to determ inethepartition

function on a generalX isperhapsslightly esoteric. Butthisfunction hasa m ore down-

to-earth interpretation ifX isofthe form S � Y .In thiscase,the theory ofthe self-dual

�ve-form G 5 on X reduceson Y ,atlow energies,toatheory ofa�ve-form on Y thatobeys

no self-duality condition.W e willcallthis�eld G 0

5.AsY isnine-dim ensional,the theory

ofG 0

5 isdualto a theory ofa four-form �eld G 0

4 on Y . G 0

4 and G 0

5 are the curvaturesof

three-form and four-form potentialsC 0

3 and C 0

4. The sam e theory on Y can be described

with eitherC 0

3 orC0

4 asthe dynam icalvariable.

(x) has,as we have seen in section 3,the following straightforward interpretation

when X isofthe form S � Y :itisa factorthatm ustbe included in the low energy path

integralfor the G 0

5 �eld on Y . To be m ore precise,ifwe regard the theory on Y as the

theory ofa �ve-form G 0

5 with ux orcharacteristicclassx = [G 0

5=2�],then perform ing the

path integralinvolvessum m ing overx. The sum isweighted with a num ber ofstandard

factors, such as an obvious factor com ing from the kinetic energy of the C 0

4 �eld. In

addition,wem ustincludein thesum overx thesign factor(x),which according to (4.2)

is

exp

�

i�

Z

Y

� [ x

�

= exp

�1

2i

Z

� ^ G 5

�

: (4:3)

On the other hand,ifwe represent the theory on Y by a four-form G 0

4 with ux or

characteristicclassu = [G 0

4=2�],then perform ing thepath integralinvolvessum m ing over

u.The sign factorwe m ustinclude in the path integralisin thiscase (a[ u)(since the

relation between G 5 and G0

4 isthattheappropriatepartofG 5 isa^ G 4),and wewillnot

determ ine (a[ u).In addition to thisin generalunknown sign factor,the path integral

over G 0

4 has another interesting e�ect,which arises by duality from (4.3).5 As we have

discussed in section 3,and as was explained from a path integralpoint ofview in [37],

a phase factor on one side is converted by duality into a shift on the other side. In the

presentinstance,since the phase in (4.3)is 1

2� (tim esG 0

5),itisconverted by duality into

a shiftin the periodsofG 0

4 by thatam ount:forany four-cycle U 2 Y ,

Z

U

G 0

4

2�=1

2

Z

U

� m od Z: (4:4)

5And,conversely to whatwe are aboutto say,the undeterm ined sign factor(a [ u)in the

G0

4 theory willby duality,ifitisnottrivial,induce a shiftin the periodsofG0

5.

15

The last form ula is essentially the result that we need. The problem posed in the

introduction was to understand,starting with the quantum m echanics ofthe Type IIB

self-dual�ve-form G 5,the factthatforany four-cycle U in a Type IIA spacetim e X ,

Z

U

G 4

2�=1

2

Z

U

� m od Z; (4:5)

wherehereG 4 istheTypeIIA four-form .IfX = S � Y with S a circle,then com ponents

of� with an index tangentto S vanish topologically (since � isa pullback from Y ),and

the interesting case of(4.5) is the case with U a four-cycle in Y . Hence the interesting

partofG 4 isthe partwith allindicestangentto Y . In the T-duality between Type IIB

and Type IIA on S � Y ,thispartofG 4 isrelated to the partofG 5 ofthe form a[ G 4,

with a a generatorofH 1(S;Z).So therelevantpartofG 4 isthesam easG0

4,and (4.4)is

equivalentto the desired relation (4.5).

In the nextsubsection,we willjustify the crucialform ula (4.2).Then in section 4.3,

we willpropose a new description of(x) in K -theory which m ay be m ore usefulfor

understanding dualitiesand theroleoftorsion.

4.2. Evaluation Of(x)

Firstwerecallfrom [31]thede�nition of(x)forageneralX .W ework on Z = S 0� X

whereS0isacirclewith aNeveu-Schwarzspin structure(thatis,S0isaspin boundary).W e

�x a generatora0ofH 1(S0;Z)= Z.Forx 2 H 5(X ;Z),wesetz = a0[ x 2 H 6(S0� X ;Z).

Now,ifW isany twelve-dim ensionalspin m anifold with boundary Z overwhich zextends,6

we set

h(x)=

Z

W

z[ z (4:6)

and (x)= (�1)h(x).For(x)to bewell-de�ned,h(x)m ustbeindependentm odulo 2 of

thechoiceofW .Thisisso becausefora closed twelve-dim ensionalspin m anifold W (that

is,onewhoseboundary vanishes),R

Wz[ z iseven forany z 2 H 6(W ;Z).(A proofofthis

assertion iscited in a footnotein section 4 of[31].)

W e could calculate m ore conveniently ifwe did not have to require W to be a spin

m anifold. However,ifwe try to use the de�nition (4.6)withoutrequiring W to be spin,

h(x) would not be well-de�ned m odulo 2 becauseR

Wz [ z is not necessarily even on a

6 How to generalize the discussion ifW doesnotexistisdiscussed in [31].W e also give m ore

generalde�nitionsof(x)below and in section 4.3.

16

generaltwelve-m anifold W . There is, however, an analogous quantity that is even in

general;itisR

W(z[ z+ v[ z),where v isthe (six-dim ensional)W u classofW .v can be

expressed in term sofStie�el-W hitney classes;forourpurposes,wecan assum e thatW is

orientable,in which case the relation isv = w2 [ w4.Thus,we are tem pted to generalize

the de�nition ofh(x)to

h(x)=

Z

W

(z[ z+ w2 [ w4 [ z) (4:7)

where W isnow required only to be oriented.(To m ake sense ofthe second integral,z is

reduced m od 2 and theintegralisunderstood in term softhecup productand integration

in m od 2 cohom ology.)

Som e care isneeded here. Though the righthand side of(4.7)isindeed even ifthe

boundary ofW vanishes, som e subtlety enters in de�ning the integralwhen W has a

nonzero boundary. An integralsuch as(4.7)isnota topologicalinvarianton a m anifold

with boundary unlesstheclassthatisbeing integrated istrivialized on theboundary;and

even ifitis,the integraldepends on the choice ofa trivialization on the boundary. (At

the levelofdi�erentialform s,thisstatem entm eansthatan integralR

W�,where � isa

twelve-form ,is not necessarily invariant under � ! �+ d� if� is nonvanishing on the

boundary.) In the case of(4.7),ifwe understand z near the boundary Z ofW to be a

pullback from Z,then z[ z vanishesneartheboundary fordim ensionalreasons,and this

trivialization isnatural.W eneed m orecare with theterm w2 [ w4 [ z.

As z and w4 are both in generalnonzero near the boundary,the only reason that

w2 [ w4 [ z vanishesneartheboundary isthatw2 does,thatis,theboundary m anifold Z

isspin. A trivialization ofw2 nearthe boundary isa choice ofspin structure on Z,and

hence we willhave to use the spin structure ofZ in de�ning the integralR

Ww2 [ w4 [ z

even though at�rstsighttheintegralappearsnotto depend on a choiceofspin structure.

A ratherdown-to-earth way to build in the spin structure ofZ isto restrictto the case

thatW isa Spincm anifold,with a Spin

cstructure thatextendsthe spin structure on Z.

TheSpincstructureon W determ inesan integrallift7 � ofw 2(W )thatissupported away

from the boundary ofW ;to be m ore precise,itdeterm ines an elem ent � ofthe relative

cohom ology group H 2(W ;@W ;Z)thatreducesto w2(W )m od 2.M oreover,ifW and W 0

aretwo Spincm anifoldswith (oppositely oriented)boundary Z and Spin

cstructuresthat

7 A Spincstructure determ inesa Spin

cbundle thatisinform ally S L

1=2where S isthe spin

bundle and L isa line bundle with c1(L)congruentto w 2(W )m od 2.IfW isnotspin,neitherS

norL1=2

existsseparately,butS L1=2

and L do.� isde�ned asc1(L).

17

extend the spin structure ofZ,then upon gluing together W and W 0 to m ake a closed

twelve-dim ensionalSpincm anifold W ,thecorresponding classes� and �0gluetogetherto

an integrallift� ofw 2(W ). In fact,� isderived from the Spincstructure on W thatis

obtained by gluing those on W and W 0. (G luing � and �0 to m ake an integrallift� of

w2(W )would notwork forarbitrary integrallifts� and �0 ofw2(W )and w2(W0);thatis

why itisim portantto derive� and �0 from Spincstructuresthatextend thatofZ.)

Finally,wegetourm oregeneralde�nition ofh(x):

h(x)=

Z

W

(z[ z+ � [ w 4 [ z): (4:8)

Thisiswell-de�ned m od 2becauseitiseven foraclosed twelve-dim ensionalSpincm anifold

W .

Evaluation For X = S � Y

W e areready to com pute forthe case thatX = S � Y ,with the supersym m etric (or

non-bounding)spin structure on S.W ehave Z = S0� X = S0� S � Y .

W e want to evaluate h(x)where x isan elem ent ofH 5(Y ;Z). W e have z = a0[ x.

To com pute h(x),we should write Z asa boundary ofa Spincm anifold W overwhich z

extends.W e could try to takeW = D 0� S � Y ,where D 0 isa two-dim ensionaldisc with

boundary S0. Thisisnotconvenientbecause a0 doesnotextend overD 0. Instead,we let

D bea discwith boundary S,and setW = S0� D � Y .Thespin structureofS doesnot

extend overD asa spin structure,butitextendsasa Spincstructure with

Z

D

� = 1: (4:9)

Asz isapullback from S0� Y ,itextendsoverW = S0� D � Y assuch apullback.Now we

can evaluate(4.8).On dim ensionalgrounds,sincez ispulled back from S0� Y ,z[ z = 0.

So weneed only considertheintegraloverS0� D � Y of� [ w 4 [ z = � [ w 4 [ a0[ x.The

integraliseasily donebecauseallfactorsarepullbacksfrom oneofthefactorsin S0� D � Y

(a0 from S0,� from D ,and the othersfrom Y ).Using (4.9)andR

S 0 a0= 1,we get

h(x)=

Z

Y

w4 [ x =

Z

Y

� [ x; (4:10)

wherethetwo expressionsareequivalentbecause on thespin m anifold Y ,� isan integral

liftofw4.Thisistheprom ised form ula (4.2).

18

4.3. K -Theory De�nition Of(x)

W e haveperform ed thiscom putation in a fram ework [31]in which (x)isde�ned as

a function on m iddle-dim ensionalcohom ology ofTypeIIB.Fortwo reasons,itseem sthat

the de�nition should be reform ulated in K -theory:

(1)In view ofT-dualitieswhich relate the Ram ond-Ram ond (RR)form sofdi�erent

dim ensions,and relate Type IIB to Type IIA,it seem s unnaturalto have a specialfor-

m alism which only appliesto them iddle-dim ensionalRR form forTypeIIB,and doesnot

apply atallfor Type IIA.Ifwe de�ne (x)in K -theory,thiswillautom atically include

alloftheRR form sofalleven orallodd dim ension,and m ay givea T-dualform alism .

(2) In view ofwhat we now know about the RR �elds,it seem s unlikely that one

can correctly take into accountthe torsion partofthe RR uxeswithoutusing K -theory

instead ofcohom ology.

Therestofthissection isdevoted toan attem pttogivea K -theory de�nition of(x).

ForTypeIIA atthelevelofdi�erentialform s,thetotalRR �eld G = G 0+ G 2+ G 4+ :::

is a sum ofdi�erentialform s ofalleven orders. For Type IIB,one has instead a sum

G = G 1 + G 3 + :::ofdi�erentialform sofallodd orders.In passing to K -theory,we will

assum e thatforType IIA,the RR ux should be regarded asan elem entx 2 K (X ).For

TypeIIB,itshould be regarded asan elem entx 2 K 1(X ).

W e will�rst de�ne a Z 2-valued function (x)= (�1)h(x) for x 2 K (x),thatis,for

TypeIIA.W ewant

(x + y)= (x)(y)(�1) (x;y); (4:11)

where (x;y)should be an integer-valued bilinear form on K (X ) that generalizes the in-

tersection pairing on cohom ology. M oreover,we want(x;y)= �(y;x),so that(x)can

be used to de�ne a line bundle on a torus K (X ;R =Z)=K (X ;Z)(by analogy with what

isdone forthe m iddle-dim ensionalcohom ology in [31]). A suitable de�nition isgiven by

index theory.Forany w 2 K (x),let

i(w)=

Z

X

bA(X )ch(w) (4:12)

bethe index ofthe Diracoperatorwith valuesin w.In ten-dim ensions,theonly term sin

ch(w)thatcontributeareterm sofdegree 4k+ 2 forsom eintegerk.These term sareodd

underw ! w (com plex conjugation ofthebundle)so

i(w)= �i(w): (4:13)

19

Then we set

(x;y)= i(x y); (4:14)

which obeys(x;y)= �(y;x)by virtueof(4.13).Thispairing vanishesifx ory istorsion;

itcan be proved thaton K (X )m od torsion,itisunim odular.

There is one m ore thing we should know about index theory in ten dim ensions. If

w is a realbundle,then i(w) = 0 because of(4.13). But there is nonetheless a natural

invariant ofw that can be de�ned using index theory. This is the \m od 2 index," the

num ber ofpositive chirality zero m odes ofthe Dirac operator with values in w,m odulo

two [38].W e willcallthisj(w).There isin generalno elem entary form ula forj(w).But

ifthe com plexi�cation ofw isofthe form x � x forsom ecom plex bundle x,then

j(w)= i(x)m od 2: (4:15)

In fact,i(x) = n+ (x)� n� (x),where n+ (x) and n� (x) are respectively the num ber of

positive and negative chirality zero m odes with values in x. Since in ten dim ensions,

com plex conjugation reversesthe chirality,we have n� (x)= n+ (x),so m odulo 2 we have

i(x)= n+ (x)+ n+ (x)= n+ (w)= j(w).

W e now can de�ne h(x),and hence (x),forTypeIIA.W e sim ply set

h(x)= j(x x): (4:16)

W em ustverify (4.11).Ifz = x� y,then z z = x x� y y� w,with w = x y� y x.

So

h(x + y)= j(z z)= j(x x)+ j(y y)+ j(w)

= h(x)+ h(y)+ i(x y)= h(x)+ h(y)+ (x;y);(4:17)

asrequired.

W ealso wanttheanalogousde�nition forTypeIIB.In thiscase,wewantto de�ne a

suitable function (x)forx 2 K 1(X ).W e interpretK 1(X )as eK (X � S1),the subsetof

K (X � S1)consisting ofelem entsthataretrivialifrestricted to X .Forx;y 2 K 1(X ),we

have x y 2 K 2(X )= eK (X � S1 � S1),and we de�ne

(x;y)=

Z

X �S 1�S 1

bA(X � S1 � S

1)ch(x y): (4:18)

Thisinteger-valued function again obeys(x;y)= �(y;x):

20

Now we want to de�ne (x). Here there is a slightsubtlety. The elem ent x x of

eK (X � S1 � S1)is replaced by its com plex conjugate ifone exchanges the two S1’s. In

addition,itistrivialifrestricted to X � S1 � p orX � p� S1,with p a pointin oneofthe

S1’s.Thesepropertiesensurethatx x can beinterpreted asan elem entofK R(X � S2),

where the realinvolution used in de�ning K R isa re ection ofonecoordinateofS 2.(By

collapsing S1 � p and p� S1,one m apsS1 � S1 to S2;the m ap thatexchanges the two

factorsofS1� S1 becom esare ection ofonecoordinatein S2.) By theperiodicity theorem

ofK R theory [39],K R(X � S2) is the sam e as K O (X ). So x x m aps to an elem ent

w 2 K O (X ),and wede�ne h(x)= j(w).The proofof(4.11)isratherasbefore.

5. System atic A nalysis For M 5-B rane

In thissection,we willcarry outan analysisofthe otherproblem m entioned in the

introduction { therelation oftheM 5-braneto theD 4-brane{ analogousto whatwehave

seen in section 4 for Type IIA/IIB.The discussion willproceed in the following stages:

�rst we willsum m arize results; then we willcom pute by hand; then we willplace the

com putation m oresystem atically in the fram ework of[31].

5.1.Outline

LetV betheworldvolum eofan M 5-branein an M -theory spacetim eM .In general,

V isoriented,butperhapsnotspin.

The subtle partofthe quantum m echanicsofthe M 5-brane isto quantize the chiral

two-form ,which hasa characteristicclassx 2 H 3(V ;Z).Thegeneralfram ework fordoing

so isanalogousto whatwesum m arized in thelastsection.Roughly speaking,onede�nes

a Z2-valued function (x)= (�1)h(x) on H 3(V ;Z),obeying theusualrelation

(x + y)= (x)(y)(�1) (x;y): (5:1)

This enables one to construct a theta function that determ ines the partition function of

the chiraltwo-form .8

In general,there isno elem entary form ula for (x). However,forthe case that the

M 5-brane can be related to a D 4-brane,there is such a form ula,in part. This is the

8 Thisdescription om itsa twistthatwe recallin section 5.2.

21

case thatV = S � R,with S a circle with supersym m etric spin structure and R a �ve-

m anifold.In thiscase,wewilljustify thefollowingassertion about(x):ifx isan elem ent

ofH 3(R;Z),then

h(x)=

Z

R

w2(R)[ x: (5:2)

Here to m ake sense ofthis integral,x should be reduced m od 2,and the integralis un-

derstood asan intersection num berin m od 2 cohom ology.To fully determ ine (x)(with

the help of(5.1)),we would also need to com pute (a[ w)fora a generatorofH 1(S;Z)

and w 2 H 2(R;Z).Itdoesnotseem thatthereisa form ula for(a[ w)aselem entary as

(5.2).

In general,the physicalapplication of(x)israthersubtle. But(asin the case we

considered in section 4),theinterpretation of(x)ism orestraightforward when V = S� R.

In thiscase,thechiraltwo-form on V reduceson R to an ordinary two-form �eld B 2 with

�eld strength T3 = dB 2 and characteristicclassx = [T3=2�],or(by duality)to a one-form

�eld B 1 with two-form �eld strength T2 = dB 1 and characteristicclassv = [T2=2�].In the

description by a two-form �eld,the evaluation ofthe path integralincludesa sum m ation

overx in which one m ustinclude the sign factor(x). Thisfactorcan be understood as

com ing from a term in the Lagrangian

i�

Z

R

w2 [ x: (5:3)

In thedualdescription by aone-form �eld,theevaluation ofthepath integralincludes

asum m ation overv.In evaluatingthissum ,oneincludesasign factor(a[v)forwhich we

willnotobtain an explicitgeneralform ula.In addition (asin thecaseconsidered in section

4),theinteraction (5.3)in thetwo-form description isdualin theone-form description to

a shiftin the periodsofT2.The dualof(5.3)isa shifted quantization law,

Z

U

T2

2�=1

2

Z

U

w2 m od Z: (5:4)

The shift m eans thatB 1,whose curvature isT2,isnot a \U (1)gauge �eld," but rather

de�nes a Spincstructure on R. (Reciprocally,the sign factor (a [ v) willin general

determ ine a shiftin the periodsofT3.)

Since R m ightnotbe Spinc,som ething ism issing in the discussion so far. There is

an im portant di�erence between (5.2) and the analogous form ula h(x) =R

R� [ x that

we m etin section 4. As� isan integralcohom ology class,the integralR

R� [ x vanishes

22

ifx is torsion;that is why torsion was not very im portant in section 4. However,w2 is

a Z2-valued cohom ology class,andRw2 [ x can perfectly wellbe non-zero fortorsion x.

W e willshow m om entarily thatprecisely when R isnotSpinc,there isa torsion classx0

with (x 0) = �1. It follows (since (x;x0) = 0 for allx,given that x0 is torsion) that

(x + x 0) = �(x) for allx. In determ ining the partition function ofthe M 5-brane,

the factor(x)isthe only factorthatisnotinvariantunderx ! x + x 0. (Forexam ple,

since x0 is torsion,the ordinary kinetic energy ofthe two-form �eld does not receive a

contribution from x0.) The contributions to the partition function from x and x + x0

willtherefore cancelin pairs,and the partition function ofthe M 5-brane vanishes. This

vanishing cannotbelifted by inserting localoperators(which do notdetecta attwo-form

�eld with characteristicclassx0),and so should beunderstood asa sortofglobalanom aly.

Existenceofthisanom aly givesan M 5-braneexplanation ofthefactthatin TypeIIA,the

D 4-brane world-volum e should beSpinc.

The existence ofx0 when R is not Spincfollows from som e basic facts in algebraic

topology.The cup productgivesa m ap

H 2(R;U (1))� H 3(R;Z)! H 5(R;U (1))= U (1) (5:5)

which by Poincar�e and Pontryagin duality isa perfectpairing. The \perfectness" m eans

thatevery hom om orphism H 3(R;Z)! U (1)isx !R

R� [ x forsom e � 2 H2(R;U (1)),

and every hom om orphism H 2(R;U (1))! U (1)is� !R

R� [ x forsom ex 2 H 3(R;Z).If

onerestrictsthepairing in (5.5)to thetorsion subgroup H 3tors(R;U (1)),then onegetsan

analogousperfectpairing

H2(R;U (1))� H 3

tors(R;Z)! U (1): (5:6)

Here H2(R;U (1)) is the group ofcom ponents ofH 2(R;U (1)) (in other words,it is the

quotientofH 2(R;U (1))bytheconnected com ponentcontainingtheidentity).Theform ula

h =R

Rw2 [ x0 isequivalentto =

R

Ri(w2)[ x0 where i:Z2 ! U (1)isthe em bedding

ofZ2 into U (1). So perfectness of(5.6)m eans thata torsion class x0 with (x 0)= �1

existsprecisely ifi(w2)isnotin theidentity com ponentofH2(R;U (1)).Now considerthe

com m utativediagram

0 ! Z2�! Z ! Z2 ! 0

# # 1

2#i

0 ! Z ! R ! U (1) ! 0

(5:7)

23

where the �rsthorizontalm ap in the top row ism ultiplication by 2,the otherhorizontal

m apsareobviousinclusionsand reductions,the�rstverticalm ap istheidentity,thesecond

verticalline ism ultiplication by 1=2,and the lastisi. Let� :H 2(R;Z2)! H 3(R;Z)be

the Bockstein derived from the �rst row,and let �0 :H 2(R;U (1)) ! H 3(R;Z) be the

Bockstein derived from the second. The condition that R is not Spincis �(w2) 6= 0;in

fact,W 3(R)= �(w2)isthe obstruction to Spincstructure. The condition thati(w2)not

bein the identity com ponentofH 2(R;U (1))isthat�0(i(w2))6= 0.Com m utativity ofthe

above diagram im plies that �0 = i�. So i(w2) is not in the identity com ponent,and a

torsion x0 with (x 0)= �1 exists,ifand only ifW 3(R)6= 0 and R isnotSpinc.

Generalizations

Thisdiscussion ofa globalanom aly isnotlim ited to the case thatV = S � R.M ore

generally,theM 5-braneisanom alouswheneverthereisatorsion classx0 with (x 0)= �1.

However,itishard in generalto givea criterion forexistence ofx0.

Iwillnow brie y suggest how these anom alies can be rem oved by turning on back-

ground �elds. In the discussion so far,we have taken the Neveu-Schwarz three-form �eld

H ofType IIA,and the corresponding M -theory four-form �eld G ,to be topologically

trivial.Naively,theclassicalequationsdT2 = H and dT = G (whereT2 isthetwo-form on

a D 4-braneand T istheself-dualthree-form on an M 5-brane)im ply thatH and G should

be trivialwhen restricted to the D 4-and M 5-brane world-volum es.However,taking into

accountthe globalanom alies,thegeneralstatem entforTypeIIA is[1,4]

H jR = W 3(R); (5:8)

whereH jR isshorthand fortherestriction toR ofthecharacteristicclassofH .Theanalog

ofthiscondition forthe M 5-brane should apparently be the following.Underthe perfect

pairing

H3(V ;U (1))� H 3

tors(V ;Z)! U (1) (5:9)

analogous to the one considered above,the function x0 ! (x 0) (for x0 torsion) corre-

spondsto an elem ent� 2 H3(V ;U (1)).The generalstatem entaboutthe restriction ofG

to V should apparently be

G jV = �0(�); (5:10)

whereasabove�0istheBockstein.Thisreducesto (5.8)in theappropriatesituation,and

Isuspectthatitholdsin general.

24

5.2.DirectCom putation

Letusnextattem ptto directly im itatethecom putation in section 4.To begin with,

we assum e thatV isspin.

Forx 2 H 3(V ;Z),we wantto de�ne a suitable Z 2-valued function (x)= (�1)h(x).

W eletZ = S0� V (with S0 a circle)and setz = a0[ x with a0 a generatorofH 1(S0;Z).9

Then,assum ing that Z is the boundary ofan eight-dim ensionalspin m anifold W over

which z extends,one istem pted to seth(x)=R

Wz[ z. Thisisnotwell-de�ned m odulo

2,becausein generalfora closed eight-dim ensionalspin m anifold W ,R

Wz[ z isnoteven.

The quantity which is always even for a closed eight-dim ensionalspin m anifold with a

given z 2 H 4(W ;Z)isR

W(z[ z+ � [ z)(where � isthe integralcharacteristicclasswith

2� = p1(W )),so we set

h(x)=

Z

W

(z[ z+ � [ z): (5:11)

Hereweneed,asin theanalogousdiscussion in section 4,tom akesenseoftheintegralR

W�[zon them anifold-with-boundary W .Thisintegralneedssom eexplanation,because

in generalneither � nor z vanishes on the boundary ofW . The approach taken in [31]

wasasfollows.If(5.11)were well-de�ned purely topologically,we would use the function

(x) to quantize the torus T = H 3(V ;R )=H 3(V ;Z) that param etrizes at three-form

�eldsC on V m od gaugetransform ations.The � [ z term in (5.11)m eansthatthe torus

thatwe can naturally quantize isnotT butthe torusT 0 thatparam etrizes,up to gauge

transform ations,C -�eldsofcurvature�=2.(T isisom orphictoT 0,by them ap C ! C + C0

where C0 isany C -�eld ofcurvature �=2,butthere isno canonicalisom orphism between

T and T 0.) A heuristic way to explain the shift from T to T 0 is that z ! z � �=2

elim inatesthez[ � term in (5.11);form oreinform ation,see[31].An alternativeapproach

to understanding theintegralin (5.11)(described to m eby M .Hopkinsand I.M .Singer)

9Thefollowingcom putation hasavery sim ilarstructuretotheonein section 4,although afew

detailsaredi�erent.To try to bring outtheanalogy,and hopefully withoutcausing confusion,we

willusesom eofthenotation ofsection 4 forobjectsthatplay theanalogousrolehere.Theseven-

m anifold Z isanalogousto the eleven-m anifold called Z in section 4;likewise,the eight-m anifold

W ofboundary Z willbe analogous to the twelve-m anifold called W in section 4. Sim ilarly,we

willuse the nam esS0;a

0;x,and z forobjectsthatplay an analogousrole to objectsofthe sam e

nam e in section 4.

25

isasfollows.The� classofa seven-dim ensionalspin m anifold such asZ isalwayseven.10

Since we only wantto de�ne h(x)m odulo 2,we can interpretthe integralR

W� [ z asan

integralin m od 2 cohom ology,replacing � and z by theirm od 2 reductions� and z.Since

� vanisheswhen restricted to the boundary ofW ,we can pick a trivialization ofit;once

such a trivialization ispicked,theintegralR

W�[ z m akessense.Therelation between the

two approachesisthata trivialization of� m od 2 givesa way ofidentifying T and T 0.

Thedetailsin thelastparagraph willnotplay a m ajorrolein thepresentpaper.The

reason isthat,with V = S � R,wewillcom pute(x)only forx 2 H 3(R;Z).Thism eans

thaton Z = S0� V = S0� S � R,both � and z = a0[ x are pullbacksfrom S0� R. In

trivializing � m od 2 on Z,we can restrict ourselves to consider only trivializationsthat

arepulled back from R,and thechoiceofsuch a trivialization doesnota�ecttheintegralR

W�[z.Atthelevelofdi�erentialform s,thislaststatem entm eansthatunder� ! �+ d ,

R

W� [ z changesby

R

S 0�S�R [ z,which vanishesfor and z both being pullbacksfrom

S0� R.Hence there isa com pletely canonical(x)forthe x we willconsider,and thisis

whatwe willevaluate.

Just as in section 4.2,it is inconvenient to calculate with W required to be a spin

m anifold.W e can readily generalizethe discussion to perm itV and W to beSpincm ani-

folds,notnecessarily spin,asfollows.A Spincm anifold W (with a chosen Spin

cstructure)

hasa two-dim ensionalclass� 2 H 2(W ;Z),which reducesm od 2 to w2(W ).In addition,

on such a m anifold p1 � �2 isdivisibleby 2,and thereisan integralcharacteristicclass�

such that11 2� = p1 � �2. M oreover,forany x 2 H 4(W ;Z),R

W(x [ x + � [ x)isalways

even.12 So we can evaluate(5.11)forany Spincm anifold W ,with � asjustde�ned.

10 The intersection form ofthe eight-m anifold B = S1� V iseven,so the relation

R

B(x [ x +

�[ x)�= 0 m odulo 2 forallx 2 H4(B ;Z)reducesto

R

B�[ x �= 0 m odulo 2 forallx.Thisim plies

that� isdivisible by 2.

11 M ore generally,any realoriented vectorbundle E with w 2(E )= 0 hasan integralcharacter-

isticclass� with 2�(E )= p1(E ).IfW isSpinc,letJ be a realtwo-plane bundle with Eulerclass

�,and letE = TW � J (with TW being the tangentbundle to W ).Then w 2(E )= 0,and �(E )

isthe desired classwith 2� = p1(E )= p1(TW )� �2.

12 Thiscan be proved by generalizing the proofgiven in section 4 of[5](see eqn.(4.7)),where

W wasassum ed to be spin.LetJ be a realtwo-plane bundle overW with Eulerclass�,and let

N be the direct sum ofJ with a trivialrank three bundle. Let K be a twelve-m anifold that is

the unitsphere bundle in N ;K isspin. Let� :K ! W be the projection,letx be any elem ent

ofH4(W ;Z),and letu be an elem entofH

4(K ;Z)with ��(u)= 1 and u [ u = 0.(Such a u can

be constructed asthe Poincar�e dualofa section of�.) Consider,as in [5],an E 8 bundle B over

26

W e willnow consider h(x) for V = S � R. W e assum e �rst that R is Spinc. W e

give V a Spincstructure thatisthe productofthe supersym m etric (orunbounding)spin

structure on S with the given Spincstructure on R. W e setZ = S0� V = S0� S � R.

Supposethatx 2 H 3(R;Z).Then asin section 4.2,Z istheboundary ofa Spincm anifold

W = S0� D � R,where D isa disc ofboundary S;and z = a0[ x extendsoverW asa

pullback from S0� R.TheSpincstructureon W istheproductofa spin structureon S0,

thegiven Spincstructureon R with two-dim ensionalclass�R ,and a Spin

cstructureon D

with a two-dim ensionalclass�D such thatR

D�D = 1.(Thereason forthelaststatem ent

is the sam e as in section 4.2: the supersym m etric spin structure on S does not extend

overD asa spin structure,butitextendsasa Spincstructure with

R

D�D = 1.) W ehave

p1(W )= p1(R)and �(W )= �D + �R ;also,�D [ �D = 0 sinceD istwo-dim ensional.W e

can com putethe� classofW :�(W )= (p1(W )� �(W )2)=2= �(R)� �D [ �R .Itfollows

that

h(x)=

Z

W

(z[ z+ �(w)[ z)=

Z

W

(z[ z+ �(R)[ z� �D [ �R [ z): (5:12)

On therighthand side,only theterm �D [ �R [ z contributestotheintegral,astheothers

arepullbacksfrom S0� R.Using z = a0[ x,withR

S 0 a0= 1,and

R

D�D = 1,we get

h(x)= �

Z

R

�R [ x: (5:13)

Since �R iscongruentto w2(R)m od 2,thisisequivalentto the prom ised form ula (5.2).

So farwehaveassum ed thatV isSpinc.Otherwise,the� classisno longeravailable,

butwe stillhave the W u classv in m od 2 cohom ology,withR

W(x [ x + v[ x)even. In

eightdim ensions,

v = w 22 + w4: (5:14)

So the de�nition ofh(x)should be

h(x)=

Z

W

�z[ z+ (w4 + w 2

2)[ z�: (5:15)

K with characteristicclassu+ ��(x).Ifi(B )istheindex ofthe D iracoperatoron K with values

in B (in the adjoint representation),then i(B ) is even (because B is realand K has dim ension

ofthe form 8k + 4).Evaluation ofi(B )via the index theorem leads,asin [5](and using the fact

that�(K )= ��(�(W ))where�(W )isde�ned asin the lastfootnote using the Spin

cstructure of

W ),to i(B )=R

W(x [ x + �(W )[ x),and so thisexpression iseven.

27

Here we have given the m ostnaturaltopologicalde�nition. In section 5.3,we willverify

thatitisequivalentto thephysics-based de�nition in [31].

In the m eantim e,we can use (5.15)to show that(5.2)istrue forallV = S � R and

x 2 H 3(R;Z),whether or not R is Spinc. For this,we note that it follows from (5.14)

thatifR istheboundary ofan oriented m anifold eR overwhich x extends,then h(x)= 0.

Forin thiscase,wecan setW = S0� S � eR,and theintegralde�ning h(x)vanishesasx;

w2,and w4 are allpullbacksfrom S0� eR. Thisbordism property can be used to reduce

to the case thatR isSpinc.Indeed,we can always�nd an oriented six-m anifold eR whose

boundary isR � R 1� R 2 (them inussignskeep track oftheorientations),wherex extends

over eR and vanishes on R 2,and R 1 is Spinc.13 The bordism property im pliesthath(x)

isthesam ewhethercom puted on S � R orS � R 1 (R 2 doesnotcontribute asx vanishes

on R 2). As R 1 is Spinc,we can use our previous result: h(x) =

R

R 1

w2 [ x. Since the

characteristic classw2(R)autom atically extendsover eR,one hasR

Rw2 [ x =

R

R 1

w2 [ x.

Hence h(x)=R

Rw2 [ x whetherornotR isSpin

c.

W e could have m ade a m uch m ore extensive use ofbordism in the present paper.

Indeed, we could have used the fact that Spin

c

5 (K (Z;2)) = Z,generated byR� [ x,

to show that (5.2) is the only nonzero bordism -invariant form ula for h(x) in the Spinc

case,whereupon we could deduce from the exam ple considered in section 2 that(5.2)is

correct. W e sim ilarly could use the fact that eSpin

9 (K (Z;5))= Z,generated byR� [ x,

plusinvarianceunderbordism ,to reducethecom putation in section 4.2to thespecialcase

considered in section 2. This would give short cuts to the desired results,but we have

chosen instead to base our com putations on a better understanding ofthe form alism in

[31].

13 Theprecisem athem aticalstatem enthereisthat 5(K (Z;3)),thebordism group oforiented

�ve-m anifoldsequipped with a three-dim ensionalcohom ology classx,isZ 2 � Z 2,a com plete set

ofinvariantsbeingRw 2 [ x and

Rw 2 [ w 3.(Thisstatem entand analogousonescited in thenext

paragraph were provided by R.Stong,along with proofs.) So forthe bordism group,we can pick

two generators R0

1 and R0

2,whereRw 2 [ x is nonzero on R

0

1 and zero on R0

2,andRw 2 [ w 3 is

nonzero on R0

2 and zero on R0

1.M oreover,one can pick R0

1 to be Spinc,and one can assum e that

x vanisheson R0

2.The factthatR0

1 and R0

2 generate the bordism group m eansthatR � R 1 � R 2

isa boundary,where the R i are asin the textand each R i isequalto R0

i orem pty,depending on

the valuesofthe invariantsofR .

28

5.3.Com parison To PhysicalDe�nition

Itrem ainsto com pare the obvioustopologicalde�nition (5.15)to the physics-based

form alism in [31].Thefullphysicalsetup forthisproblem dependson detailsthatwehave

so farom itted. The M 5-brane worldvolum e V isem bedded in an eleven-m anifold M . V

isorientable (butnotnecessarily spin),and M isspin.LetN be the norm albundle to V

in M .Thecondition forM to bespin is

w1(N )= 0; w2(N )= w2(V ): (5:16)

Also,the EulerclassofN vanishes(orequivalently,asN isofodd rank,w5(N )= 0),for

reasonsexplained in section 5 of[5].Anotherpartofthe data isthe four-form �eld G of

M -theory.ItisoftheformG

2�=�(M )

2+ g; (5:17)

where g isan integralclass.M oreover,ifU isa sm allfour-sphere linking V in M ,then

Z

U

g = 1; (5:18)

since the �vebrane hasunitcharge.

LetP bethesubm anifold ofM consisting ofallpointsa distance� from V ,forsom e

very sm all�. P isa four-sphere bundle overV . Let� :P ! V be the projection. (5.18)

isthe statem entthat

��(g)= 1: (5:19)

Thisuniquely determ inesg m odulo g ! g+ ��(y)fory 2 H 4(V ;Z).Notethat��(g[ g)

isinvariantm od 2 undersuch a transform ation ofg.Hence,itsm od 2 reduction doesnot

depend on the choice ofg.In fact,

��(g[ g)�= w4(N )m od 2: (5:20)

Toprovethis,sincethelefthand sideisindependentofthechoiceofg m odulo 2,itsu�ces

to considerthecase thatg isthe Poincar�edualto a section s of�.(Such a section exists

atleastoverthe�ve-skeleton ofV ,sincetheEulerclassofN iszero,and a choiceofs on

the �ve-skeleton su�cesforevaluating the four-dim ensionalclasson the lefthand side of

(5.20).) Choiceofsuch a section splitsN asN = O � N 0whereO isa rank onetrivialreal

bundle (consisting ofm ultiplesofs)and N 0 isa rank fourbundle.g[ g isPoincar�e dual

29

to theintersection classs\ s.Ifweregard sasa codim ension foursubm anifold ofP ,then

itsnorm albundleisN 0,so s\ sisdualto therestriction to softheEulerclass�(N 0),and

hence��(g[ g)= ��(g[ �(N0))= ��(g)[ �(N

0)= �(N 0).But(forany SO (4)bundleN 0)

�(N 0)iscongruentto w4(N0)m od 2,and with N = O � N 0,we have w4(N )= w4(N

0).

Thisjusti�esthe assertion in (5.20).

Pick aclassx 2 H 3(V ;Z).W ewillnow restatethede�nition of(x)= (�1)h(x) given

in [31].Letz = a0[ x 2 H 4(S0� V ;Z),with a0a generatorofH 1(S0;Z).Let eZ = S0� P ,

where S0 isa circle with Neveu-Schwarz spin structure. Thus, eZ isa four-sphere bundle

overZ = S0� V ;we write e� forthe projection e� :eZ ! Z. And de�ne w 2 H 4(eZ;Z)by

w = e��(z)+ g.Letnow fW be a twelve-dim ensionalspin m anifold with boundary eZ over

which w extends.Such a fW alwaysexists[40].The de�nition in [31]can be stated

h(x)=1

3

Z

eW

�

w �1

2�

��

(w �1

2�)2 �

1

8(p2 � �2)

�

� (w ! 0): (5:21)

Here the m eaning ofthe lastterm is thatone should subtract the sam e expression with

w replaced by 0. E 8 index theory isused to prove thath(x)isintegraland independent

m odulo 2 ofthe choice offW and ofthe extension ofw. The fact that the class that is

integrated in (5.21)isnotcanonically trivialnearthe boundary m eansthatthe function

(x)enablesusto quantize notthe space H 3(V ;U (1))of atthree-form �eldson V ,but

a shifted version ofit.

Thede�nition ofh(x)justgiven isratherabstract.Forcom putation,itisconvenient

to m ake som e sim plifying assum ptionsthatare actually ratherm ild in practice.Suppose

thatS0� V isthe boundary ofan oriented eight-m anifold W overwhich N extends(asa

rank �ve bundle obeying w 1(N )= 0,w2(N )= w2(W ),and w5(N )= 0). Let fW be the

unitsphere bundle in N ;the conditionson N ensure thatfW isspin,and itsboundary is

eZ = S0� P . Suppose further that z = a0[ x extends over W ,and that g extends over

fW . Then,setting w = ��(z)+ g,we can sim plify (5.21)by integrating overthe �bersof

� :fW ! W .W e get

h(x)=

Z

W

�

z[ z� z[ �(fW )+ z[ ��(g[ g)

�

: (5:22)

(W ehavedropped term sthatvanish ifx = 0;they in factvanish m od 2 using thefactthat

the integralin (5.21) is even ifevaluated on a closed twelve-dim ensionalspin m anifold,

and thefactthattheintegrand vanishesnearthe boundary ifx = 0.)

30

To clarify this further,we would like to express the m od 2 reduction of��(fW )+

��(g[ g)in term sofquantitiesde�ned juston W .Forthis,wenote�rstthat(forany spin

m anifold fW )�(fW )iscongruentm od 2tow4(fW ).Stably,thetangentbundlesTfW and TW

offW and W arerelated by TfW = TW � N .So since w1(W )= w1(N )= 0 and w2(N )=

w2(W ),wehavew4(fW )= w4(W )+ w2(W )w2(N )+ w4(N )= w4(W )+ w2(W )2 + w4(N ).

Usingalso(5.20),welearn that��(fW )+ ��(g[g)iscongruentm od 2tow4(W )+ w2(W )2,

so that(5.22)isequivalentto

h(x)=

Z

W

(z[ z+ w4(W )[ z+ w2(W )[ w2(W )[ z): (5:23)

Thisisthe form ula thatwe guessed on purely form algrounds toward the end ofsection

5.2. W hat we have gained is an understanding ofhow this form ula is related to eleven-

dim ensionalphysics.

Thiswork wassupported in partby NSF G rantPHY-9513835 and the Caltech Dis-

covery Fund.Iam gratefultoM .J.Hopkinsand I.M .Singerfornum erousexplanationsof

theirviewpointaboutthe �vebrane action,aswellasotherm atters.In addition,Iwould

liketo thank G .M oorefordiscussionsand suggestionsaboutthem anuscript,R.Stong for

helpfulcorrespondence,and E.Diaconescu,D.Freed,and A.Kapustin forcom m entsand

questions.

31

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34