A BIVARIANT CHERN CHARACTER FOR FAMILIES OF SPECTRAL TRIPLES

Denis Perrot

Description

A BIVARIANT CHERN CHARACTER FOR FAMILIES OF SPECTRAL TRIPLES Denis PERROT SISSA, via Beirut 2-4, 34014 Trieste, Italy September 7, 2004 Abstract In this paper we construct a bivariant Chern character defined on “families of spectral triples”. Such families should be viewed as a version of unbounded Kasparov bimodules adapted to the category of bornological algebras. The Chern character then takes its values in the bivariant entire cyclic cohomology of Meyer. The basic idea is to work within Quillen's algebra cochains formalism, and construct the Chern character from the exponential of the curvature of a superconnection, leading to a heat kernel regularization of traces. The obtained formula is a bivariant generalization of the JLO cocycle. Keywords: Bivariant entire cyclic cohomology, bornological algebras. 1 Introduction Recall that according to Connes [6], a noncommutative space is described by a spectral triple (A,H, D), where H is a separable Hilbert space, A an asso- ciative algebra represented by bounded operators on H, and D is a self-adjoint unbounded (Dirac) operator with compact resolvent, such that the commutator [D, a] is densely defined for any a ? A and extends to a bounded operator. The triple (A,H, D) carries a nontrivial homological information as a K-homology class of A.

• unbounded bimodules


• vector space


• triples over


• algebras


• any bounded


• bivariant entire


• cyclic cohomology


• algebras provide


• cohomology theory

Sujets

• Perrot
• Vector space
• Algebra
• Cyclic homology
• Cohomology

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 9
• Langue: English

Extrait

ABIVARIANTCHERNCHARACTERFOR
FAMILIESOFSPECTRALTRIPLES

DenisPERROT
SISSA,viaBeirut2-4,34014Trieste,Italy
[e-mail]
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