A BIVARIANT CHERN CHARACTER FOR FAMILIES OF SPECTRAL TRIPLES

ABIVARIANTCHERNCHARACTERFOR
FAMILIESOFSPECTRALTRIPLES

DenisPERROT
SISSA,viaBeirut2-4,34014Trieste,Italy
perrot@fm.sissa.it
September7,2004

Abstract
InthispaperweconstructabivariantCherncharacterdefinedon
“familiesofspectraltriples”.Suchfamiliesshouldbeviewedasaversion
ofunboundedKasparovbimodulesadaptedtothecategoryofbornological
algebras.TheCherncharacterthentakesitsvaluesinthebivariantentire
cycliccohomologyofMeyer.ThebasicideaistoworkwithinQuillen’s
algebracochainsformalism,andconstructtheCherncharacterfromthe
exponentialofthecurvatureofasuperconnection,leadingtoaheatkernel
regularizationoftraces.Theobtainedformulaisabivariantgeneralization
oftheJLOcocycle.
Keywords:
Bivariantentirecycliccohomology,bornologicalalgebras.

1Introduction
RecallthataccordingtoConnes[6],anoncommutativespaceisdescribedby
aspectraltriple(
A
,
H
,D
),where
H
isaseparableHilbertspace,
A
anasso-
ciativealgebrarepresentedbyboundedoperatorson
H
,and
D
isaself-adjoint
unbounded(Dirac)operatorwithcompactresolvent,suchthatthecommutator
[
D,a
]isdenselydefinedforany
a
∈A
andextendstoaboundedoperator.The
triple(
A
,
H
,D
)carriesanontrivialhomologicalinformationasa
K
-homology
classof
A
.ThemajormotivationleadingConnestointroduceperiodiccyclic
cohomology[5]isthatthelatteristhenaturalreceptacleforaCherncharacter
definedon
finitelysummable
representativesof
K
-homology.Thisfinitenesscon-
ditionwasremovedlaterandreplacedbytheweakerconditionof
θ
-summability,
i.e.theheatkernelexp(
−
tD
2
)associatedtothelaplacianoftheDiracoperator
hastobetrace-classforany
t>
0[7].Inthatcase,thealgebra
A
hastobe
endowedwithanormandtheCherncharacterofthespectraltripleisexpressed
asan
infinite-dimensionalcocycle
intheentirecycliccohomology
HE
∗
(
A
).Ex-
cepttheoriginalconstructionofConnes,oneoftheinterestingexplicitformulas
forsuchaCherncharacterisprovidedbytheso-calledJLOcocycle[21].Here
theheatkernelplaystheroleofa
regulator
inthealgebraofoperatorson
H
,and
theJLOformulaincorporatesthedataofthespectraltripleinarathersimple
way.ThisledConnesandMoscovicitousethepowerfulmachineryofasymp-
toticexpansionsoftheheatkernel,givingriseto
local
expressionsextendingthe
1

classicalindextheoremsofAtiyah-Singertoveryinterestingnon-commutative
situations[9,10].
InthispaperwewanttogeneralizetheconstructionofaCherncharacterto
familiesofspectraltriples“overanoncommutativespace”describedbyasecond
associativealgebra
B
.Inthecontextof
C
∗
-algebras,suchobjectscorrespond
totheunboundedversionofKasparov’sbivariant
K
-theory[4].Inthispicture,
anelementofthegroup
KK
(
A
,
B
)isrepresentedbyatriple(
E
,ρ,D
),where
E
isanHilbert
B
-module.
D
shouldbeviewedasafamilyofDiracoperatorsover
B
,actingbyunboundedendomorphismson
E
,and
ρ
isarepresentationof
A
asboundedendomorphismsof
E
commutingwith
D
moduloboundedendomor-
phisms.Intheparticularcase
B
=
C
,thisdescriptionjustreducestospectral
triplesover
A
.TheconstructionofageneralbivariantCherncharacterasa
transformationfromanalgebraicversionof
KK
(
A
,
B
)(for
A
and
B
notneces-
sarily
C
∗
-algebras)toabivariantcycliccohomologyhasalreadybeenconsidered
byseveralauthors.ForexampleNistor[25,26]constructedabivariantChern
characterfor
p
-summablequasihomomorphisms[12],withvaluesintheJones-
Kasselbivariantcycliccohomologygroups.CuntzandQuillenalsoconstructed
abivariantCherncharacterundersomesummabilityassumptions,withvalues
intheirowndescriptionofthebivariantperiodiccyclictheory[15,16].Onthe
otherhand,Puschniggconstructedawell-behavedcycliccohomologytheoryfor
C
∗
-algebras,namelythe
localcycliccohomology
[29,30].Upongeneralizationof
apreviousworkofCuntz[13],thelocalcyclictheoryappearstobethesuitable
targetforacompletelygeneralbivariantCherncharacter(withoutsummabil-
ityassumptions)definedofKasparov’s
K
-theory.However,theexistenceand
propertiesofsuchconstructionsareoftenbasedonexcisionincycliccohomology
andtheuniversalpropertiesofbivariant
K
-theory.Byconsideringunbounded
bimoduleswewillfollowadifferentway,involvingheatkernelregularization
inthespiritoftheJLOcocycle,keepinginmindthatweareinterestedin
ex-
plicitformulas
forabivariantCherncharacterincorporatingthedata
ρ
and
D
.
Ourmotivationmainlycomesfromthepotentialapplicationstomathematical
physics,especiallyquantumfieldtheoryandstring/branetheory,wheresuch
objectsarisenaturally:
•
Theheatkernelmethodadmitsafunctionalintegralrepresentation.The
quantitiesunderinvestigationthencorrespondtoexpectationvaluesofobserv-
ablescorrespondingtosomequantum-mechanicalsystem.Thiswasfirstused
byAlvarez-Gaume´andWittenintheirstudyofmixed-gravitationalanomalies
[1],andledtotheasymptoticsymbolcalculusofGetzler[2].
•
ThebasicideaofintroducingaheatkernelregularizationofCherncharacters
inclassicaldifferentialgeometryisduetoQuillen[31].Bismutthensuccesfully
appliedthismethodinhisapproachoftheAtiyah-Singerindextheoremfor
familiesofellipticoperatorsonsubmersions[3].Itisworthmentioningthat
Bismutalsousesastochasticrepresentationoftheheatkernel.
•
TheBismut-Quillenapproachisessentialfortheanalyticandtopological
understandingofanomalies(bothchiralandgravitational)inquantumfield
theory[27,28].AbivariantCherncharacterdesignedinanequivariantsetting
mayshedsomelightontheinterplaybetweenBRScohomologyandtherecently
discoveredcycliccohomologyofHopfalgebras[10,11].
•
Twisted
K
-theoryand
K
-homologyrecentlyappearedinthephysicsliterature
2

throughtheclassificationof
D
-branes[23,34].Thisalsofallsintothescopeof
abivariantCherncharacter.
Firstwehavetoconsidertherightcategoryofalgebras.Forourpurpose,
itturnsoutthat
bornologicalassociativealgebras
areexactlywhatweneed.
Theseareassociativealgebrasendowedwithanadditionalstructuredescribing
thenotionofa
boundedsubset
.Completebornologicalalgebrasprovidethe
generalframeworkforentirecycliccohomology.Thistheoryhasbeendeveloped
indetailbyMeyerin[24].Theinterestingfeatureofthebivariantentirecyclic
cohomologyisthatitcontainsinfinite-dimensionalcocyclesandthuscanbeused
asthereceptacleofabivariantCherncharacterforourfamiliesofspectraltriples
carryingsomepropertiesof
θ
-summability.Giventwocompletebornological
algebras
A
and
B
,wewillconsiderthe
Z
2
-gradedsemigroupΨ
∗
(
A
,
B
),
∗
=
0
,
1,of
unbounded
A
-
B
-bimodules
.ThelatterisanadaptationofKasparov’s
unboundedbimodulestotherealmofbornologicalalgebras.Inourgeometric
picture,suchabimodulerepresentsafamilyofspectraltriplesoverthenon-
commutativespace
B
.OuraimistoconstructanexplicitformulaforaChern
characterdefinedonthesubsemigroupof
θ
-summablebimodules,
ch:Ψ
∗
θ
(
A
,
B
e
)
→
HE
∗
(
A
,
B
)
,
∗
=0
,
1
,
(1)
carryingsuitablepropertiesofadditivity,differentiablehomotopyinvariance
andfunctoriality.Here
HE
∗
(
A
,
B
)isthebivariantentirecycliccohomology
of
A
and
B
,and
B
e
istheunitalizationof
B
.Onthetechnicalside,wewill
useboththe
X
-complexdescriptionofcycliccohomologyduetoCuntz-Quillen
[15,16],andtheusual(
b,B
)-complexofConnes.The
X
-complexisusefulfor
someconceptualexplanationsoftheabstractpropertiesofcyclic(co)homology.
Givenacompletebornologicalalgebra
A
,itsentirecyclichomologyiscomputed
bythesupercomplex[24]
X
(
TA
):
TA
⇄
Ω
1
TA
♮
,
(2)
where
TA
isthe
analytictensoralgebraof
A
,obtainedbyacertainbornological
completionofthetensoralgebraover
A
,andΩ
1
TA
♮
=Ω
1
TA
/
[
TA
,
Ω
1
TA
]
isthecommutatorquotientspaceoftheuniversalone-formsover
TA
.This
meansthattheentirecyclichomologyof
A
iscompletelydescribedthrough
thehomologicalpropertiesofitsanalytictensoralgebraindimension0and
1.Furthermore,takingtheanalytictensoralgebraof
TA
isharmless:indeed
X
(
TA
)and
X
(
TTA
)arehomotopicallyequivalentcomplexes.Inotherwords,
entirecyclichomologydoesnotdistinguishbetweenacompletebornological
algebraanditssuccessivenestedanalytictensoralgebras.Thisisaparticular
caseoftheanalyticversion[24]ofGoodwillie’stheorem[18].Thisresultisa
keypointofourbiva