The geometry of determinant line bundles in noncommutative geometry

0.24 MB·English

by  Partha Sarathi Chakraborty

THE GEOMETRY OF DETERMINANT LINE BUNDLES
IN NONCOMMUTATIVE GEOMETRY
PARTHASARATHICHAKRABORTYANDVARGHESEMATHAI
Abstract. This paper is concerned with the study of the geometry of de-
terminantlinebundlesassociatedtofamiliesofspectraltriplesparametrized
bythemodulispaceofgaugeequivalentclassesofHermitianconnections on
9 a Hermitian finite projective module. We illustrate our results with some
0 examplesthatariseinnoncommutative geometry.
0
2

n 2000 Mathematics Subject Classification: 58B34 (primary), 46L87, 58G26 (secondary)
a
J

0 Introduction
2
Inthemid1990s,ConnesandMoscovici[4]formulatedandprovedafarreachinglocal

] index theorem for spectral triples and introduced the correct definition of dimension in
h the noncommutative setting, where it is no longer an integer, but rather a subset of C
t
- which is called the dimension spectrum. This paper aims to understand the stability
p of the dimension spectrum for families of spectral triples, and its implications on the
e
existence of geometric structures on the determinant line bundle, such as the Quillen
h
metric and the determinant section.
[
Thatis,weareconcernedwiththestudyofthegeometryofdeterminantlinebundles

3 associated to families of spectral triples (A,H,D) parametrized by the moduli space
v of gauge equivalent classes of Hermitian connections on a Hermitian finite projective
2
module. Recallthatspectraltriples(A,H,D)wereintroducedbyConnes[2],asdefining
3 a noncommutative manifolds, where A is a separable, unital C∗-algebra acting on a
2
separable Hilbert space H, D an unboundedself-adjoint operator on H such that D has
3
compact resolvent, and [D,a] is a bounded operator on H for all a∈A.
.
4 Given a spectral triple (A,H,D), a finite projective module E over A, a Hermitian
0 structure on E and any Hermitian connection ∇ on E, a basic stability result implies
8
that (End (E),E ⊗ H,D ) is again a spectral triple. The space of all Hermitian
A A E,∇
0
connections on E is an affine space C , and the gauge group G is defined to be a Lie
: E
v subgroup of the group Aut (E) of invertible elements in End (E), such that G acts
A A
i smoothly and freely on C . Analogous to the classical case we define the determinant
X E
linebundleLoftheindexbundleforthisfamily ofspectraltriples. Thisisalinebundle
r
on themoduli space C /G of gauge equivalent classes of Hermitian connections on E.
a E
In order to state the hypotheses required to define the Quillen metric [12] and the
determinantsectionofL,weneedtoutilisethenotionsofregularityandsimpledimension
spectrum introduced in [4]. More precisely, we require the spectral triple (A,H,D) to
be regularwithsimple dimensionspectrumandzeroisnotinthe dimensionspectrum.
ThesearepreciselythesameassumptionsmadebyConnesandChamseddine[3]intheir
Keywords and phrases. regularity,spectraltriples,determinantlinebundles,indextheorem
forfamilies,connections, gaugetransformations,dimensionspectrum.
1
2 PARTHASARATHICHAKRABORTYANDVARGHESEMATHAI
workoninnerfluctuationsofspectralactions,exceptthatwedonotneedtheassumption
that zero is not in the spectrum of D, cf. §4.3. In this context, Higson [9] also treats
the case when zero is in the spectrum of D, but our approach differs from his. Another
technicalresultprovedhereisthestabilitypropertyforregularspectraltriples(A,H,D),
which saysthat (End (E),E⊗ H,D ) is again aregular spectral triple with simple
A A E,∇
dimension spectrum for any Hermitian structureon E and any Hermitian connection ∇
on E. For the application of these constructions to a mathematical understanding of
anomalies, that is, the nonpreservation of a symmetry of the classical action by the full
quantumaction in a gauge field theory,see [1, 6].
The last section works out an explicit calculation of the Quillen metric and determi-
nantsectionofthedeterminantlinebundleoftheindexbundleforthefamilyofspectral
triplesonthenoncommutativetorusparametrizedbythemodulispaceofflatYang-Mills
connectionsonafreemoduleofrankonewhichwerestudiedbyConnes-Rieffel[5]. These
areexpressedintermsofthethetaandetafunctionsonthemodulispacewhichisatorus.
In[11],Perrot hasstudied aK-theoreticindextheorem for families ofspectral triples
parametrized by the moduli space of gauge equivalent classes of Hermitian connections
on E. He makes the restrictive assumption that the gauge group G be contained in
the unitary group of A, which is unnecessary in our context here, and he also does not
constructthedeterminantlinebundleoftheindexbundleanditsgeometry,whichisthe
main study in this paper.
1. Preliminaries
In this section, we recall the construction of the determinant line on the Banach
manifold of all bounded Fredholm operators acting between separable Hilbert spaces.
This motivates the constructions used later in the paper.
Recall that a Fredholm operator T : H −→ H between two infinite dimensional
0 1
Hilbert spaces H ,H , is a bounded linear operator such that dim(ker(T)) < ∞ and
0 1
dim(coker(T)) < ∞. This implies in particular that Im(T) is a closed subspace of H .
1
Let F = F(H ,H ) denote the space of all Fredholm operators between two infinite
0 1
dimensionalHilbertspacesH ,H . ItfollowsfromAtkinson’stheoremthatF isanopen
0 1
subset of the Banach space B(H ,H ), of all bounded linear operators between the two
0 1
Hilbert spaces, which establishes in particular that F is a Banach manifold modeled on
theBanach space B(H ,H ). It has countablymany connected componentslabelled by,
0 1
index:π (F)→∼= Z, where for T ∈F,
0
index(T)=dim(ker(T))−dim(coker(T)).
We want to review the construction of a smooth line bundle DET → F, called the
determinant line bundle, such that DET = Λmax(ker(T))∗ ⊗ Λmax(coker(T)). The
T
obvious definition does not work since dim(ker(T)) jumps as T varies smoothly.
This problem was essentially fixed in [12]. For the convenience of the reader we
elaborate on his solution. Let Gr denote the space of all finite dimensional subspaces
fin
of H . Consider the open cover {U } of F, where F ∈ Gr and U = {T ∈ F :
1 F fin F
Im(T)+F =H }. ForT ∈U , consider theexact sequenceof finitedimensional vector
1 F
spaces,
(1) 0→ker(T)→T−1F →T F →coker(T)→0.
Sinceindexisconstantonsmoothfamilies,andtherankofF isfixedonU ,thereforethe
F
rankofT−1F isconstantonsmoothfamiliesinU . SoEF →U definedbyEF =T−1F,
F F T
is a smooth vector bundle. The virtual vector bundle INDEXF → U is defined to be
F
thepair (EF,F), where F denotes thetrivial vectorbundleover U with fibreF.
F
THE GEOMETRY OF DETERMINANT LINE BUNDLES IN NCG 3
UsingtheinnerproductsonH , i=0,1,thesequenceinequation(1)splits,ker(T)⊕
i
F ∼=T−1F ⊕coker(T),therefore
Λmax(ker(T))∗⊗Λmax(coker(T))∼=Λmax(T−1F)∗⊗ΛmaxF.
The determinant line bundle, DETF → U , is defined as the smooth line bundle,
F
DETF =det((INDEXF)∗), i.e. DETF =Λmax(EF)∗⊗ΛmaxF.
Supposethat T ∈U ∩U . Then we havetheexact sequences
E F
0 //ker(T) ι1 //T−1F ⊕(E\F) T⊕1 //F +E //coker(T) //0
= ϕ = =
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //ker(T) ι2 //T−1E⊕(F \E) T⊕1 //F +E //coker(T) //0,
where ϕ is uniquely defined so as to make the diagram commute. Here by (E\F) we
mean thequotient,E/(E∩F) etc.. By theFive Lemma, themap ϕ is an isomorphism,
and therefore, DETF = Λmax(EF)∗ ⊗ Λmax(E \F)∗ ⊗ Λmax(F + E) and DETE =
Λmax(EE)∗⊗Λmax(F\E)∗⊗Λmax(F+E)arenaturallyisomorphicviaΛmaxϕ⊗1over
U ∩U . Here we have used the fact that Λmax(V)∗⊗Λmax(V) is canonically trivial
E F
for any vector bundle V. By the clutching construction, it follows that DET defines a
smooth line bundleover theunion U ∪U .
E F
To show that DET defines a smooth line bundle over the whole of F, we need to
showthat ontriple overlapsU ∩U ∩U ,therearenaturalisomorphisms between the
E F G
determinant line bundles DETE, DETF and DETG. The proof is similar to the case of
doubleoverlaps. Suppose that T ∈U ∩U ∩U . Then we havetheexact sequences
E F G
0 //ker(T) ι1 //T−1G⊕(E+F \G) T⊕1 //E+F +G //coker(T) //0
= ϕ1 = =
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //ker(T) ι1 //T−1F ⊕(E+G\F) T⊕1 //E+F +G //coker(T) //0
= ϕ2 = =
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //ker(T) ι2 //T−1E⊕(F +G\E) T⊕1 //E+F +G //coker(T) //0,
whereϕ , j =1,2,isuniquelydefinedsoastomakethediagram commute. BytheFive
j
Lemma, the map ϕ , j =1,2, is an isomorphism, and therefore,
j
DETG=Λmax(EG)∗⊗Λmax(E+F \G)∗⊗Λmax(E+F +G)
and
DETF =Λmax(EF)∗⊗Λmax(E+G\F)∗⊗Λmax(E+F +G)
are naturally isomorphic via Λmaxϕ ⊗1 over U ∩U ∩U , where we have used the
1 E F G
fact that Λmax(V)∗⊗Λmax(V) iscanonically trivial for anyvectorbundle V. Similarly,
DETF =Λmax(EF)∗⊗Λmax(E+G\F)∗⊗Λmax(E+F +G)
and
DETE =Λmax(EE)∗⊗Λmax(F +G\E)∗⊗Λmax(E+F +G)
are naturally isomorphic via Λmaxϕ ⊗1 over U ∩U ∩U . Therefore DETG,DETF
2 E F G
and DETE are canonically identified on the overlaps U ∩U ∩U . By the clutching
E F G
construction,itfollows thatDETdefinesaconsistent smoothlinebundleovertheunion
4 PARTHASARATHICHAKRABORTYANDVARGHESEMATHAI
U ∪U ∪U . Since E,F and G are arbitrary finite dimensional subspaces of H , it
E F G 1
follows that DET defines a smooth line bundleon the Banach manifold F.
ItisafactthatF ishomotopyequivalenttoZ×BU(∞),sothatbyBottperiodicity,
π2j(F) ∼= Z and π2j+1(F) = {0} for j ≥ 0. So by Hurewicz’s theorem, H2(F0,Z) ∼= Z,
whereF istheconnectedcomponentofFconsistingofFredholmoperatorsofindexequal
0
tozero. ItisafactthatthefirstChernclass,c1(DET)isthegeneratorofH2(F0,Z)∼=Z.
2. Stability of spectral triples coupled to a Hermitian finite projective
module with Hermitian connection
ThenotionofaspectraltriplewasintroducedbyConnes[2]asthecriteria defininga
noncommutative spin geometry. A slightly weakened version, dropping the postulate of
theexistenceofarealstructure,wasgivenbyMoscovici[10]whichwassuitedtodescribe
more general Poincar´e dual pairs of algebras, and in particular noncommutative spinc
geometries.
LetAbeaseparable,unitalC∗-algebraactingonaseparableHilbertspaceH. LetD
beanunboundedself-adjoint operatoronH. LetAbeasmooth unitalsubalgebraofA.
Then (A,H,D) is said to be a spectral triple if D has compact resolvent, and [D,a] is a
boundedoperatoronHforalla∈A. (A,H,D)issaidtobeevenifthereisaself-adjoint
involutiononHwithrespect towhichtheaction ofAisevenandD isanoddoperator.
Otherwise the spectral triple is said tobe odd. Webegin by considering theeven case.
Let E be a finite projective (right-)module over A. A Hermitian structure on E is a
sesquilinear map (,):E×E −→A satisfying the following conditions:
(1) (ξa,ηb)=a∗(ξ,η)b,∀ξ,η∈E and ∀a,b∈A;
(2) (ξ,ξ)≥0;
(3) E is self-dual with respect to (,).
Consider the special case of the free A-module E = Aq which has a canonical Her-
0
mitian structuregiven by (ξ,η)= qξ∗η , ∀ξ=(ξ ,...,ξ ),η=(η ,...,η )∈E .
1 j j 1 q 1 q 0
LetE beafiniteprojective(right)moduleoverA. IfwewriteE asadirectsummand
E = eAq of a free module E , whPere e ∈ M (A) is a self adjoint projection. Then E
0 q
hasaHermitianstructurewhichisobtainedbyrestrictingtheHermitianstructureonE
0
defined above, to E. That is, every finite projective module E over A has a Hermitian
structure.
Consider the A-bimodule of 1-forms on A, Ω1(A) := { a [D,b ];a ,b ∈A}. A
D j j j j
Hermitian connection on E is a C-linear map ∇ : E → E ⊗ Ω1 (A) satisfying the
PA D
following:
(1) ∇(ξa)=∇(ξ)a+ξ⊗da, ∀ξ∈E,a∈A, (Leibnitzproperty)
(2) (ξ,∇η)−(∇ξ,η) = d(ξ,η), ∀ξ,η ∈ E; where da = [D,a], ∀a ∈ A, (Hermitian
property)
If ∇(ξ)= ξ ⊗ω , with ξ ∈E,ω ∈Ω1 (A), then (∇ξ,η)= ω∗(ξ ,η).
j j j j D j j
AnexampleofaHermitianconnectionistheGrassmannianconnection∇ onE =eAq
0
P P
given by ∇ (ξ) = edξ, where dξ = (dξ ,...dξ ). Also, any two Hermitian connections
0 1 q
differbyaselfadjointelementofEnd (E)⊗ Ω1 (A). Thatis,thespaceofallHermitian
A A D
connectionsonE isanaffinespaceC withassociatedvectorspaceEnd (E)⊗ Ω1 (A).
E A A D
Given a Hermitian finite projective module E over A, we can form the Hilbert space
E⊗ H, by completing the algebraic tensor product with respect to the inner product
A
hξ ⊗η ,ξ ⊗η i=hη ,(ξ ,ξ )η i∀ξ ∈E,η ∈H.
1 1 2 2 1 1 2 2 j j
Given such a pair (E,∇), one can define a twisted operator D on the Hilbert
E,∇
space E ⊗ H by setting D (ξ ⊗η) = ξ ⊗D(η)+∇(ξ)η, ∀ξ ∈ E,η ∈ H. Here, if
A E,∇
∇(ξ)= ξ ⊗ω , then ∇(ξ)η= ξ ⊗ω (η)∈E⊗ H.
j j j j A
P P
THE GEOMETRY OF DETERMINANT LINE BUNDLES IN NCG 5
Thewe havethefollowing.
Lemma 2.1. Let T be an unbounded self-adjoint operator on a Hilbert space H and
suppose that there is an operator S ∈I, where I denotes either the ideal of all compact
operatorsonHoraSchattenidealonH,suchthat(iI+T)S−I ∈I. Then(iI+T)−1∈I.
Moreover, (iI+R+T)S−I ∈I for any bounded self-adjoint operator R, and therefore
(iI+R+T)−1∈I.
Proof. To prove this, notethat
(iI+T)−1=S−K(iI+T)−1
whereK =(iI+T)S−I ∈I. Therefore therighthandsideisintheidealI asclaimed.
SinceRS ∈Iforanyboundedself-adjointoperatorR,itfollowsthat(iI+R+T)S−I ∈I,
and therefore (iI+R+T)−1∈I as claimed.
(cid:3)
Proposition2.2(StabilityofSpectralTriplesI). Let(A,H,D)beaspectraltriple,E be
afiniteprojectiveA-modulewithHermitianstructureandHermitianconnection∇. Then
(End (E),E⊗ H,D ) is also a spectral triple. Moreover if (A,H,D) isp-summable
A A E,∇
then so is (End (E),E⊗ H,D ).
A A E,∇
Proof. First observe that for all natural numbers N, (M (C)⊗A,CN ⊗H,1⊗D) is a
N
spectraltriple,whereCN⊗H∼=AN⊗AH. SotheresultistruewhenE isafreemodule
and with the trivial connection. For simplicity of notation, denote by D the operator
I ⊗D. An arbitrary connection on the free module E is of the form D+ a [D,a′],
i i i
where the term a [D,a′], a ,a′ ∈ A, is a bounded operator which we denote by R.
i i i i i P
Such perturbations of D will be referred as inner fluctuations. The condition that the
P
connection is Hermitian implies that R is self-adjoint. By Lemma 2.1, it follows that
(A,H,D+R) isagain aspectral triple, which is just thestatement that spectral triples
are stable underinner fluctuations.
Next we observe there exists a projection e ∈ M (A) for some natural number N
N
such that E = eAN. We want to show that (End (E),E ⊗ H,D ) is a spectral
A A E,∇0
triple, where ∇ =e[D,e] is theGrassmann connection and we have used thesimplified
0
notation. Consider thecompact operator S =e(iI+D)−1e. Then
(2) e(iI+D)eS =e(iI+D)e(iI+D)−1e
(3) =e[(iI+D),e]e(iI+D)−1e+e
sothate(iI+D)eS−e∈K. ByLemma2.1,itfollowsthate(iI+D)−1e=(ie+eDe)−1
isacompactoperator,thatis(End (E),E⊗ H,D )isaspectraltriple. Thisalong
A A E,∇0
withthestabilityofspectraltriplesunderinnerfluctuationsasprovedaboveimpliesthe
result for arbitrary connections on E. Replacing the ideal of compact operators K by
Schatten ideals we get thesummability result. (cid:3)
3. Universal families of spectral triples and determinant line bundles
We follow a setup that is analogous to that of Freed [6], where it was done in the
classical caseoffamilies ofDiracoperatorsonasmoothmanifold. Thisnoncommutative
geometrydatawillenableustodefinethedeterminantlinebundleassociatedtoanatural
family of spectral triples.
6 PARTHASARATHICHAKRABORTYANDVARGHESEMATHAI
3.1. Noncommutative Geometry Data for determinant line bundles:
(1) A spectral triple, (A,H,D);
(2) A Hermitian finite projective moduleE overA;
(3) AgaugegroupGwhichisaLiesubgroupofthegroupAut (E)ofautomorphisms
A
ofE,suchthatG actssmoothlyandfreelyonC asfollows. For ξ∈E,∇∈C
E E
and g∈G, defineg.∇(ξ):=(g−1⊗1)∇(g.ξ);
Remarks 3.1. (1) There is an associated Z -graded Hilbertian bundleE⊗ H over
2 A
C /G.
E
(2) Therefore one obtains a family of spectral triples, (EndA(E),E⊗AH,DE,∇y),
y∈C /G. Forease of notation, denoteD =D . Then D can beviewed as
E y E,∇y
an odd degree bundlemap D:E⊗ H→E⊗ H.
A A
(3) Under our assumptions, the quotient C /G is a smooth manifold. If B is a
E
compact smooth finite dimensional submanifold of C /G, then all of the non-
E
commutativedata restricts to B.
(4) In the case of the spectral triple for a spin manifold, this is just the data for a
family of twisted Dirac operators.
TheconstructionofthedeterminantlinebundleduetoQuillen[12]isbrieflyadapted
in our context in the rest of the section. Consider the G-equivariant family of spec-
tral triples {(End (E),E⊗ H,D ): ∇∈C }. The G-equivariant family of finite-
A A E,∇ E
dimensional spaces kerD+ ⊂ E ⊗ H+ and kerD− ⊂ E ⊗ H− defines a virtual
E,∇ A E,∇ A
bundleIndex(DE,∇E) overB, which is theindexbundlefor thefamily, whose fibreat ∇
is thevirtual vector space Index(DE,∇E)∇ =kerDE+,∇−kerDE−,∇.
Now D− D+ and D+ D− are self-adjoint operators, with discrete spectrum,
E,∇ E,∇ E,∇ E,∇
and spec(D− D+ ) = spec(D+ D− ). The simple argument to show this goes as
E,∇ E,∇ E,∇ E,∇
follows. If e is an eigenvector of D∗ D with eigenvalue λ, then D e is an eigen-
E,∇ E,∇ E,∇
vector of D D∗ with eigenvalue λ. Also if f is an eigenvector of D D∗ with
E,∇ E,∇ E,∇ E,∇
eigenvalueη, thenD∗ f is an eigenvectorof D∗ D with eigenvalue η. This shows
E,∇ E,∇ E,∇
that for λ 6= 0, D is an isomorphism from the λ-eigenspace of D∗ D to the
E,∇ E,∇ E,∇
λ-eigenspace of D D∗ (with inverse given by 1D∗ ).
E,∇ E,∇ λ E,∇
Let H+ denote the span of eigenvectors of D− D+ with eigenvalue < λ, and
∇,λ E,∇ E,∇
H− denote the span of eigenvectors of D+ D− with eigenvalue < λ. These are
∇,λ E,∇ E,∇
smooth vector bundles over the open subset U = {∇ ∈ B : λ 6∈ spec(D− D+ ) =
λ E,∇ E,∇
spec(D+ D− )}.
E,∇ E,∇
It is easy to show that there is an exact sequence,
0→ker(D+ )→H+ →H− →ker(D− )→0.
E,∇ ∇,λ ∇,λ E,∇
This gives rise to a canonical isomorphism,
Λmax(ker(D+ )∗)⊗Λmax(ker(D− ))∼=Λmax(H+ ∗)⊗Λmax(H− )
E,∇ E,∇ ∇,λ ∇,λ
Therefore we obtain a smooth line bundle L := Λmax(H+ ∗)⊗Λmax(H− ) over the
λ ∇,λ ∇,λ
opensetU . Ifµ>λ,thenH± =H± ⊕H± ,whereH+ denotesthespanofthe
λ ∇,µ ∇,λ ∇,λ,µ ∇,λ,µ
eigenvectors of D− D+ with eigenvalues lying in theopen interval (λ,µ) and H−
E,∇ E,∇ ∇,λ,µ
denotes the span of the eigenvectors of D+ D− with eigenvalues lying in the open
E,∇ E,∇
interval (λ,µ). Therefore ΛmaxH± ∼= ΛmaxH± ⊗ΛmaxH± . Since the restriction
∇,µ ∇,λ ∇,λ,µ
D+ : H+ → H− is an isomorphism as observed earlier, we deduce that
E,∇ H∇+,λ,µ ∇,λ,µ ∇,λ,µ
det(D˛˛E+,∇ H∇+,λ,µ) : ΛmaxH∇+,λ,µ → ΛmaxH∇−,λ,µ is also an isomorphism. Therefore on
the overla˛ps Uλ∩Uµ, there is a canonical identification of the determinant line bundles
˛
THE GEOMETRY OF DETERMINANT LINE BUNDLES IN NCG 7
L and L given by s7→s⊗det(D+ ). Since {U :λ∈Q} is an open cover of
λ µ E,∇ H∇+,λ,µ λ
B, we obtain a global determinant line˛bundle L overB associated to the G-equivariant
˛
familyofspectraltriples{(A,E⊗ H,D ): ∇∈C }. Westatethisasaproposition.
A E,∇ E
Proposition 3.2(Determinantlinebundle). Let(A,H,D)beaspectral triplesatisfying
theassumptionsinsection3.1,andBasmoothcompactsubmanifoldofC /G. Thenthere
E
is a smooth determinant line bundle L over B associated to the G-equivariant family of
spectral triples {(End (E),E⊗ H,D ): ∇∈C },whose fibre at∇∈B isnaturally
A A E,∇ E
isomorphic to Λmax(ker(D+ )∗)⊗Λmax(ker(D− )).
E,∇ E,∇
4. Stability of regular spectral triples, the Quillen metric and
determinant section
The Quillen metric on the determinant line bundle, is obtained by patching metrics
constructedontheopensetsU . Asintheclassicalcasethispatchingrequiresazetareg-
λ
ularizationofthemetricsonU . Thisinturnrequiresfurtherassumptionsonthespectral
λ
triple similar to the ones that Connes and Moscovici used. Let H∞ = Dom|D|n;
n≥1
in thissection we will assume that every a∈A maps H∞ to itself.
T
Definition 4.1. (1) Aspectraltriple(A,H,D)issaidtoberegularifAand[D,A]
iscontained in thedomain ofδk, for all k. Hereδ is thederivation a7→[|D|,a].
(2) For a regular spectral triple (A,H,D) let B denote the algebra generated by
0
γ,Sign(D), δk(a),δk([D,a]),a ∈ A,k ≥ 0. The dimension spectrum of a p-
summableregularspectraltriple(A,H,D)isthesmallestdiscretesubsetΣ⊂C
with theproperty that all thezeta functions
ζ(s,a)=Tr(a|D+P|−s), a∈B , s∈C, ℜ(s)>p
0
havemeromorphiccontinuationstoCwithpolescontainedinΣ,whereP denotes
theorthogonal projection onto thenullspace of D.
(A,H,D) is said to have simple dimension spectrum if the associated zeta
functions ζ(s,a) haveonly simple poles, for all a∈B .
0
Ournextstability result is thefollowing.
Proposition 4.2 (Stability of Regular Spectral Triples). Let (A,H,D) be a regular
spectral triple with simple dimension spectrum Σ, E be a finite projective A-module with
Hermitian structure and Hermitian connection ∇. Then (End (E), E ⊗ H, D )
A A E,∇
is also a regular spectral triple with simple dimension spectrum Σ′ contained in Σ−N.
Moreover if 0∈/ Σ, then 0∈/ Σ′.
Wedefer theproof of this proposition to §4.2 and §4.3.
4.1. Noncommutative Geometry Data for the Quillen metric and the deter-
minant section: We next modify the noncommutative geometry data given in section
3.1, that will enable us to definethe Quillen metric on the determinant line bundleand
also definethedeterminant section.
(1) Aregular spectral triple,(A,H,D)suchthatzeroisnotinthesimpledimension
spectrum;
(2) A Hermitian finite projective moduleE overA;
(3) AgaugegroupGwhichisaLiesubgroupofthegroupAut (E)ofautomorphisms
A
ofE,suchthatG actssmoothlyandfreelyonC asfollows. For ξ∈E,∇∈C
E E
and g∈G, defineg.∇(ξ):=(g−1⊗1)∇(g.ξ);
8 PARTHASARATHICHAKRABORTYANDVARGHESEMATHAI
To simplify notation, we will denote ζ(s) := ζ(s,1). Since by the hypotheses above,
0 is not in the dimension spectrum Σ of our regular spectral triple (A,H,D), it enables
us to define the derivative at zero ζ′(0). In particular, the regularized determinant
det(D∗D) = e−ζ′(0) makes sense, and is defined to be zero if 0 ∈ spec(D). Also, if
ζ (s)=Tr(χ (|D|)|D′|−s)foru>0andu∈/ spec(|D|)andwhereχ (|D|)denotes
u [u,∞] [u,∞]
the spectral projection of |D|, then there is a simple relationship between the two zeta
functions,
1
ζ(s)= +ζ (s), λ∈spec(|D|)
λs u
0<Xλ<u
Since 1 , λ ∈ spec(|D|), is an entire function, it follows that ζ (s) also has
0<λ<u λs u
an analytic continuation to C\Σ, so that in particular, the derivative at zero ζ′(0) is
P u
defined. Moreover, by the stability of regular spectral triples, Proposition 4.2, we see
thattheζ′(0,∇)andζ′(0,∇)aredefined,whereζ (s,∇)=Tr(χ (|D |)|D′ |−s)
u u [u,∞] E,∇ E,∇
foru∈/ spec(|D |)andwhereχ (|D |)denotesthespectralprojectionof|D |
E,∇ [u,∞] E,∇ E,∇
and ζ(s,∇)=ζ (s,∇).
0
Giventhegeometricdataasgivenabove,wewillproceedtofirstconstructtheQuillen
metriconthedeterminantlinebundleL. Usingthenotation ofsection 3.1,wearegiven
a Hermitian inner product on E⊗ H, which induces Hermitian metrics on the vector
A
bundlesH , since for each ∇∈C , H is a finitedimensional subspace of the Hilbert
λ E λ,∇
space E ⊗ H. Let g′ denote the induced Hermitian metric on the determinant line
A µ
bundle L over the open set U . Due to the canonical identification of the determinant
λ λ
line bundles L and L overU ∩U given by s7→s⊗det(D+ ), where µ>λ,
λ µ µ λ E,∇ H∇+,λ,µ
we see that ˛
2 ˛
g′(s,s)=g′(s,s). τ .
µ λ 0 1
λ<τ<µ,τY∈spec(DE)
Therefore the induced Hermitian metric@g′ does not define aAsmooth Hermitian metric
λ
on L, but however can be modified as in [12] to give a smooth Hermitian metric g on
L called the Quillen metric as follows. At ∇ ∈ Uλ, define gλ = gλ′.e−ζλ′(0,∇). Then for
∇∈U ∩U , since
λ µ
e−ζλ′(0,∇) =e−ζµ′(0,∇) τ ,
0 1
λ<τ<µ,τY∈spec(DE,∇)
@ A
we see that g = g on the overlap U ∩U , showing that g := g defines a smooth
λ µ λ µ L
Hermitian metric on thedeterminant line bundle L.
Proposition 4.3 (Quillen metric on the determinant line bundle). Let (A,H,D) be a
regular spectral triple with dimension spectrum not containing zero. Let B be a compact
submanifold of C /G. Then there is a smooth Hermitian metric g , called the Quillen
E L
metric) on the determinant line bundle L over B associated to the G-equivariant family
of spectral triples {(End (E),E⊗ H,D ): ∇∈C }.
A A E,∇ E
We next describe the determinant section of L over the open set U ∩ U . Let
λ 0
{ψ ,...,ψ } beabasis of eigenvectors in H+ and {ψ∗,...,ψ∗} bethedualbasis. Then
1 n λ 1 n
thedeterminant section is det(D+ )=(ψ∗∧...∧ψ∗)⊗(Dψ ∧...∧Dψ ). On the
E,∇,λ 1 N 1 N
overlap U ∩U ∩U , it is easy to see that
λ µ 0
det(D+ )=det(D+ ) τ .
E,∇,µ E,∇,λ 0 1
λ<τ<µ,τY∈spec(DE)
@ A
THE GEOMETRY OF DETERMINANT LINE BUNDLES IN NCG 9
Itfollows thatontheopensetU ,thereisasmooth sectiondet(D )ofthedeterminant
0 E
line bundleL. Therefore,
Proposition 4.4 (Determinant section of the determinant line bundle). Let (A,H,D)
be a regular spectral triple with dimension spectrum not containing zero. Let B be a
compact submanifold of C /G. Then there is a smooth determinant section det(D ) of
E E
the determinant line bundle L, over B ∩U associated to the G-equivariant family of
0
spectral triples {(End (E),E⊗ H,D ): ∇∈C }.
A A E,∇ E
4.2. Proof of Proposition 4.2. We begin by proving the following special case of
Proposition 4.2.
Proposition4.5(StabilityofRegularSpectralTriples). Let(A,H,D)bearegularspec-
traltriplewithsimpledimensionspectrum Σ,E beafiniteprojectiveA-modulewithHer-
mitian structure and Hermitian connection ∇. In addtion, we assume that 0∈/ spec(D)
and 0 ∈/ spec(D ). Then (End (E),E⊗ H, D ) is also a regular spectral triple
E,∇ A A E,∇
with simple dimension spectrum Σ′ contained in Σ−N. Moreover if 0∈/ Σ, then 0∈/ Σ′.
To prove regularity we will use Higson’s characterization of regularity. For that we
recall,
Definition 4.6. Let (A,H,D) beaspectral triple suchthat AmapsH∞ toitself. The
algebraofdifferentialoperatorsD(A,∆)isthesmallestalgebraofoperatorsonH∞closed
undertheoperation T 7→[∆,T]containing A and [D,A]. Here ∆ denotes D2.
This algebra is filtered as follows, the elements of A and [D,A] have order zero and
theoperation[∆,·]raisesorderatmostbyone. ThusD ,thespaceofoperatorsoforder
k
at most k are definedinductively as follows
D = algebra generated by A and [D,A]
0
D = D +[∆,D ]+D [∆,D ]
1 0 0 0 0
k−1
D = D +[∆,D ]+D [∆,D ]+ D D
k 0 k−1 0 k−1 j k−j
j=1
X
Definition4.7. Aspectraltriple(A,H,D)satisfiesthebasicestimateifeverydifferential
operator X ∈D thereis an ǫ>0 such that
k
kDkvk+kvk≥ǫkXvk
for all v∈H∞.
Theorem 4.8 (Higson, Theorem 4.26, [9]). Let (A,H,D) be a spectral triple such that
everya∈AmapsH∞ toitself. Thenthisspectral tripleisregular iffitsatisfies thebasic
estimate.
Proof. FornotationalconvenienceletusdenoteD byD′. Let∆′ =D′2andD(End (E),∆′)
E,∇ A
be the associated differential algebra with the filtration D′. Then note that D′ ⊆ D .
k k
Therefore (End (E),E⊗ H,D′) satisfies thebasic estimate henceregularity.
A A
Now we proceed to the stability of the dimension spectrum. This is the final part
of Proposition 4.5. We will divide this in two cases. First let us tackle the case of free
modules. Then if necessary by considering matrices with entries from A we can assume
E =A.
10 PARTHASARATHICHAKRABORTYANDVARGHESEMATHAI
Lemma 4.9. Let T ∈B and 0<ℜ(λ)<min{d∈σ(∆)}, then
(a)
n k
(λ−∆)−1T = T |D|j(λ−∆)−1−k+R (λ,T)
jk n
Xk=1Xj=0
where
k
T = 2jδ2k−j(T).
jk j!
(b) kR (λ,T)k <Cmax(|ℑλ|,1)−n/2.
n 1
C+i∞
(c)Thefunctionz7→ λ−zR (λ,T)D(λ−∆)−1dλisholomorphicforlargeenough
n
ZC−i∞
n where C is a real number separating zero from the spectrum of ∆.
Proof. (a) To prove(a) note,
(λ−∆)−1T = T(λ−∆)−1+[(λ−∆)−1,T]
= T(λ−∆)−1+(λ−∆)−1[∆,T](λ−∆)−1
= T(k)(λ−∆)−1−k+(λ−∆)−1T(n+1)(λ−∆)−n
0≤Xk≤n
Here the last equality is obtained by iterating the previous one k-times and T(k)’s are
defined inductively as T(0) = T and T(k) = [∆,T(k−1)]. For b ∈ B we also have the
following relations:
[∆,b] = |D|δ(b)+δ(b)|D|
|D|b = b|D|+δ(b)
Combining these two we get for T ∈B,
(4) T(k) =2δ(T(k−1))|D|+δ2(T(k−1))
Applyingthisrepeatedly we get
k
(5) T(k) = 2jδ2k−j(T)|D|j
j!
0≤Xj≤k
(6) =T |D|j.
jk
(b) To prove(b) note,
d
k|D|(λ−∆)−1/2k ≤ sup
|λ−d2|
d∈σ(|D|)
d2
= sup
d∈σ(|D|)s(ℜ(λ)−d2)2+ℑ(λ)2
d2
≤ sup ifℑ(λ)>1;
d∈σ(|D|)s(ℑ(λ)2(ℜ(λ)−d2)2)
1
≤
|ℜ(λ)|1/4|(1− ℜ(λ)|)
p