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
R eferences
[1] E.W itten,\Baryons and Branes In Antide Sitter Space," JHEP 9807:006 (1998),
hep-th/9805112.
[2] R. M inasian and G . M oore, \K Theory And Ram ond-Ram ond Charge," JHEP
9711:002 (1997),hep-th/9710230.
[3] E.W itten,\D -BranesAnd K Theory," JHEP 9812:019 (1998),hep-th/9810188.
[4] D. Freed and E. W itten, \Anom alies In String Theory W ith D -Branes," hep-
th/9907189.
[5] E.W itten,\On Flux Quantization In M Theory And TheE�ectiveAction,"J.G eom .
Phys.22 (1997)1,hep-th/9609122.
[6] N.M arcus and J.H.Schwarz,\Field Theories That Have No M anifestly Lorentz
InvariantForm ulation," Phys.Lett.B 115 (1982)111.
[7] W . Siegel, \M anifest Lorentz Invariance Som etim es Requires Nonlinearity," Nucl.
Phys.B 238 (184)307.
[8] P.G oddard and D.Olive,\Algebras,Lattices,and Strings" (preprint,1983),Phys.
Scripta T 15 (1987).
[9] C.Im bim bo and A.Schwim m er,\The Lagrangian Form ulation OfChiralScalars,"
Phys.Lett.B 193 (1987)455.
[10] C.M .Hull,\CovariantQuantization OfChiralBosonsAnd Anom aly Cancellation,"
Phys.Lett.B 206 (1988)234.
[11] J.M .F.Labastidaand M .Pernici,\LagrangiansForChiralBosonsAnd TheHeterotic
String," Nucl.Phys.B 306 (1988)516.
[12] L.M ezincescu and R.I.Nepom echie,\CriticalDim ensionsForChiralBosons," Phys.
Rev.D 37 (1988)3067.
[13] M .Henneaux and C.Teitelboim ,\Dynam ics OfChiral(Selfdual) p Form s," Phys.
Lett.B 206 (1988)650;F.P.Devecchiand M .Henneaux,\CovariantPath Integral
ForChiralp-Form s," Phys.Rev.D 54 (1996)1606.
[14] P.P.Srivastava,\On A G augeTheory OfSelfdualField And ItsQuantization,"Phys.
Lett.B 234 (1990)93.
[15] B.M cClain,Y.S.W u,and F.Yu,\Covariant Quantization OfChiralBosons and
O Sp(1;1j2)Sym m etry,Nucl.Phys.B 343 (1990)689.
[16] C.W otzase,\TheW ess-Zum ino Term ForChiralBosons," Phys.Rev.Lett.66 (1991)
129.
[17] I.M artin and A.Restuccia,\Duality Sym m etric Actions And CanonicalQuantiza-
tion," Phys.Lett.B 323 (1994)311.
[18] J.H.Schwarz and A.Sen,\Duality Sym m etric Actions," Nucl.Phys.B 411 (1994)
35,hep-th/9304154.
32
[19] E.Verlinde,\G lobalAspectsOfElectric-M agneticDuality,"Nucl.Phys.B 455 (1995)
211,hep-th/9506011.
[20] M .Perry and J.H.Schwarz,\Interacting ChiralG auge Fields In Six Dim ensional
Born-Infeld Theory," hep-th/9611065.
[21] J.H.Schwarz,\Coupling A Self-DualTensor To G ravity In Six Dim ensions," hep-
th/9701008.
[22] N.Berkovits,\LocalActionsW ith Electric and M agnetic Sources," hep-th/9610134,
\Super-M axwellActionsW ith M anifestDuality," hep-th/9612174.
[23] I.Bengtsson and A.Kleppe,\On Chiralp-Form s," hep-th/9609102;I.Bengtsson,
\M anifestDuality In Born-Infeld Theory," hep-th/9612174.
[24] P.Pasti,D.Sorokin,and M .Tonin,Phys.Lett.B 352 (1995) 59,Phys.Rev.D 52
(1995)R4277,\On Lorentz InvariantActionsForChiralP -Form s," hep-th/9611100,
\Covariant Action For A D = 11 Five-Brane W ith The ChiralField," Phys.Lett.
B 398 (1997) 41;I.Bandos,K.Lechner,A.Nurm agam betov,P.Pasti,D.Sorokin,
\Covariant Action For The Super-Five-Brane Of M -Theory," hep-th/9701149; G .
Dall’Agata,K.Lechner,and M .Tonin,\Action For IIB SUpergravity In Ten Di-
m ensions," hep-th/9812170.
[25] M .Aganagic,J.Park,C.Popescu,and J.Schwarz,\W orldvolum e Action For The
M -Theory Fivebrane," Nucl.Phys.B 496 (1997)191,hep-th/9701166.
[26] P.S.Howe,E.Sezgin,and P.C.W est,\CovariantField EquationsOfTheM Theory
Five-Brane,"Phys.Lett.B 399 (1997)49,hep-th/9702008,\TheSix-Dim ensionalSelf-
DualTensor," Phys.Lett.B 400 (1997)255,hep-th/9702111.
[27] Y.-G .M iao,J.-G .Zhou,and Y.-Y.Liu,\New W ay OfTheDerivation OfFirstOrder
W ess-Zum ino Term s," Phys.Lett.B 323 (1994)169;Y.-G .M iaoand H.J.W .M uller-
Kirsten,\Self-Duality OfVariousChiralBoson Actions," hep-th/9912066.
[28] A.M azyntsia,C.R.Preitschopf,and D.Sorokin,\DualActionsForChiralBosons,"
hep-th/9808049.
[29] X.Bekaert,M .Henneaux,and A.Sevrin,\Deform ations OfChiralTwo-Form s In
Six Dim ensions," hep-th/9909094,\Sym m etry-deform ing Interactions Of Chiralp-
Form s," hep-th/9912077.
[30] M .Henningson,B.E.W .Nilsson,and P.Salom onson,\Holom orphic Factorization
OfCorrelation Functions In (4k + 2)-Dim ensional(2k)-Form G auge Theory," hep-
th/9908107.
[31] E.W itten,\FivebraneE�ectiveAction In M -Theory,"J.G eom .Phys.22 (1997)103,
hep-th/9610234.
[32] L.Dolan and C.R.Nappi,\A M odularInvariant Partition Function ForThe Five-
Brane," Nucl.Phys.B 530 (1998)683,hep-th/9806016.
[33] M .Hopkinsand I.M .Singer,to appear.
33
[34] D.J.G ross,J.A.Harvey,E.M artinec,and R.Rohm ,\The HeteroticString," Phys.
Rev.Lett.54 (1985)502.
[35] D.Freed,J.A.Harvey,R.M inasian,and G .M oore,\G ravitationalAnom aly Can-
cellation For M Theory Five-Branes," Adv.Theor.M ath.Phys.2 (1998)601,hep-
th/9803205.
[36] K.Beckerand M .Becker,\Five-BraneG ravitationalAnom alies," hep-th/9911138.
[37] E.W itten,\On S Duality In Abelian G augeTheory," hep-th/9505186.
[38] M .F.Atiyah and I.M .Singer,\TheIndex OfEllipticOperators:V," Ann.M ath.93
(1971)139.
[39] M .F.Atiyah,\K -Theory And Reality," Quart.J.M ath.,Oxford (2)17 (1966)367.
[40] R.Stong,\CalculationOfspin
11 (K (Z;4),"in Uni�edStringTheories,eds.M .B.G reen
and D.J.G ross(W orld Scienti�c,1986).
34