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Noncommutative Geometry
and Geometric Analysis
on Heisenberg Manifolds
Raphael Ponge
(Ohio State University)
1
I. Heisenberg Manifolds
Definition. A Heisenberg manifold is a man-
ifold M together with a distinguished hyper-
plane H ⊂ TM .
The most important examples of Heisenberg
manifolds are:
- Foliations;
- Contact manifolds;
- Cauchy-Riemann manifolds.
2
• CR manifolds
Definition. A CR structure on an orientable
manifold M2n+1 is given by a rank n complex
subbundle T1,0 ⊂ TCM so that:
- T1,0 is integrable, i.e. [T1,0, T1,0] ⊂ T1,0.
- T1,0 ∩ T0,1 = 0, where T1,0 = T1,0.
Equivalently, H = <(T1,0⊕T0,1) has the struc-
ture of a complex vector bundle of rank n.
Examples. 1. Heisenberg group H2n+1.
2. Hypersurface M ⊂ Cn.
3. Circle bundles over complex manifolds.
3
• Motivations for studying CR manifolds:
1. Important geometric objects (thanks to
work of Cartan, Chern-Moser, Kohn, Feffer-
man, Tanaka, Webster ...).
2. Step towards a geometric index formula
for complex manifold with boundary (i.e. bound-
ary version of the Riemann-Roch theorem).
3. Step towards the study of the NCG of
Lorentzian manifolds with Fefferman metric.
4
• Contact Manifolds:
Definition. A contact structure on an ori-
entable manifold M2n+1 is given by a nonva-
nishing 1-form θ such that dθ is everywhere
nondegenerate on H = ker θ.
Examples. 1. Heisenberg group H2n+1.
2. Nondegenerate hypersurface M ⊂ Cn.
3. Boundary of symplectic manifold, e.g. co-
sphere bundle S∗M of a manifold M .
5
• Motivations for studying contact manifolds:
1. Important geometric structure (thanks to
work of Elyashberg, Gromov, Rumin ...).
2. Step towards a geometric index formula
for symplectic manifold with boundary.
6
• Tangent Lie Group Bundle:
Lemma. There exists a 2-form L : H ×H →TM/H so that
Lm(X(m), Y (m)) = [X,Y ](m) mod Hm
for sections X, Y of H near m ∈M .
Definition.The tangent Lie group bundle GM
is obtained by endowing the bundle,
(TM/H)⊕H
with the grading and product such that
t.(X0 +X ′) = t2X0 + tX ′, t ∈ R,(X0 +X ′).(Y0 + Y ′) =
X0 + Y0 +1
2L(X ′, Y ′) +X ′ + Y ′,
for sections X0, Y0 of TM/H and X ′, Y ′ of H.
7
Proposition. We have:
rkLm = 2n⇐⇒ GmM ' H2n+1 × Rd−2n,
where H2n+1 the (2n+1)-dimensional Heisen-
berg manifold.
This result justifies the terminology Heisen-
berg manifold.
Heisenberg Chart:
Definition. 1) A H-frame is a local frame
X0, X1, . . . , Xd of TM such that X1, . . . , Xdspan H.
2) A Heisenberg chart is a local chart for
M together with a H-frame on its domain.
8
Remarks (RP, arXiv ‘04). 1) We can char-
acterize the type of a Heisenberg manifold
(foliation, contact, CR) by means of the struc-
ture of its tangent Lie group bundle.
2) Using some privileged coordinates at every
point x ∈ M we can interpret GM as the
Lie group of a Lie algebras span by jets of
the vector fields X0, X1, . . . , Xd of a H-fame
around x.
3) The privileged coordinates allows us to get
a straightforward construction of the tangent
groupoid, very much close to Connes’ original
construction of the tangent groupoid of a
manifold.
9
• List of Main Geometric Operators:
- ∂b-complex and its associated Laplacian,
the Kohn Laplacian, on a CR manifold.
- Rumin’s contact complex and its associated
Laplacian on a contact manifold;
- Horizontal (sub-)Laplacian ∆H on a general
Heisenberg manifold;
- Hormander (sub-)Laplacian ∆ = −(X21 +
. . .+X2p ) +X0, where X1, . . . , Xd span H.
With the exception of the contact Laplacian
these Laplacians are sublaplacians, i.e. locally
of the form
∆ = −(X21+. . .+X2
d )+λX0+µ1X1+. . .+µdXd+ν,
for some H-frame X0, X1, . . . Xd.
10
II. Heisenberg Calculus
The Heisenberg calculus is the right pseudod-
ifferential tool to study the main differential
operators on Heisenberg manifolds.
It was independently introduced by Beals-
Greiner (‘84) and M. Taylor (‘84), extending
previous works by Dynin, Folland-Stein and
Boutet de Monvel.
11
• Differential Operator of Order m
Continuous operator P : C∞(M) → C∞(M)
locally of the form
P =∑
|α|≤maα(x)D
αx , Dx =
1
i
∂
∂x.
Associated to P is its symbol
p(x, ξ) =∑
|α|≤maα(x)ξ
α,
so that P = p(x,Dx) and we have
Pu(x) = (2π)−n∫eix.ξp(x, ξ)u(ξ)dξ.
12
• Classical Symbol of Order m:
Using this last formula we can define p(x,Dx)
for more general symbols such that
p ∼∑j≥0
pm−j, pm−j(x, λξ) = λm−jp(x, ξ),
i.e. for any N ≥ 0 as |ξ| → ∞ we have
|DαxD
βξ (p−
∑j≤J
pm−j)(x, ξ)| = O(|ξ|−N)
for J ≥ Jαβ large enough.
13
• ΨDO of Order m:
Continuous operator P : C∞c (M) → C∞(M)
locally of the form
P = p(x,Dx) +R,
where p is a classical symbol of order m and
R is a smoothing operator.
14
• Heisenberg Calculus (Local Theory)
Let U ⊂ Rd+1 be a Heisenberg chart with H-
frame X0, . . . , Xd. The Heisenberg calculus is
such that:
- X0 has order 2 and X1, . . . , Xd order 1;
- At any x ∈ U the calculus is modelized by
that of homogeneous left-invariant convolu-
tion operators on the tangent group GxU .
15
• Heisenberg Symbols of order m
Smooth functions on U × Rd+1 such that
q ∼∑j≥0
qm−j, qm−j(x, λ.ξ) = λm−jqm−j(x, ξ),
where λ.ξ = (λ2ξ0, λξ1, . . . , λξd)
Since the vector fields Xj’s are approximated
by left-invariant vector fields Xxj ’s at any point
x ∈ U we use them to quantize the Heisen-
berg symbols. Let σj(x, ξ) be the classical
symbol of 1iXj and set
σ = (σ0, . . . , σd) and X = (X1, . . . , Xd).
Then −iX = σ(x,Dx) and we quantize a Heisen-
berg symbol q(x, ξ) by letting
q(x,−iX) = q(x, σ(x,Dx)).
16
Definition.An operator Q : C∞c (U) → C∞(U)
is a ΨHDO of order m when it is of the form
Q = q(x,−iX) +R,
where q is a Heisenberg symbol of order m
and R is smoothing.
We let ΨmH(U) denote the space of ΨHDO’s
of order m.
17
• Model Operator and Symbolic Calculus
Definition.Let Q ∈ ΨmH(U). Then the model
operator of Q at x is the left-invariant ΨH on
GxU given by
Qx = qxm(−iXx),
where qxm = qm(x, .) and qm(x, ξ) is the prin-
cipal symbol of Q.
Proposition.For j = 1,2 let qj be a homoge-
neous Heisenberg symbol of degree mj. Then
there exists a homogeneous Heisenberg sym-
bol q = q1 ∗ q2 of degree m1 +m2 so that
(q1 ∗ q2)x(−iXx) = qx1(−iXx)qx2(−iX
x).
18
Remark. This convolution is local w.r.t. x,
but is neither commutative, nor microlocal
(i.e. local w.r.t. ξ).
Using the convolution for Heisenberg sym-
bols we get:
Proposition.Let Qj ∈ ΨmjH (U), j = 1,2. Then
Q = Q1Q2 is a ΨHDO of order m1 +m2 and
we have
qm1+m2= qm1 ∗ qm2 and (Q1Q2)
x = Qx1Qx2.
19
Heisenberg Calculus (Global Theory):
Proposition (Beals-Greiner, Taylor). The
class of ΨHDO′s of order m is invariant by
the action of Heisenberg diffeomorphisms.
This allows us to define ΨHDO’s of order m
on any Heisenberg manifold (M,H).
Proposition (RP ’04). To any P ∈ ΨmH(M)
we can associate:
- A model operator Px at any point x as a
left-invariant convolution operator on GxM ;
- A principal symbol as a homogeneous smooth
function on g∗M \ 0.
20
Proposition. For P ∈ ΨmH(M) TFAE:
(i) The principal symbol of P is invertible
(w.r.t. convolution for Heisenberg symbols).
(ii) P admits a parametrix Q ∈ Ψ−mH (M)
(i.e. we have QP = PQ = 1 mod Ψ−∞(M)).
Moreover, if (i) and (ii) hold then P is hy-
poelliptic, i.e. we have Sobolev estimates
‖u‖s+m/2 ≤ C(‖Pu‖s + ‖u‖s) ∀u ∈ C∞c (M).
The main geometric operators on Heisenberg
manifolds are hypoelliptic in the above sense,
except the Kohn Laplacian which has an in-
vertible principal symbol only outside forms
of some given bidegrees.
21
III. Hypoelliptic Functional Calculus
In order to be able to make use of the NCG
framework we shall now construct complex
powers of hypoelliptic ΨHDO’s.
Here we’re facing a big technical hurdle: due
to the non-microlocality of the Heisenberg
calculus we cannot carry through Seeley’s ap-
proach to the complex powers.
Instead we will rely on two new approaches:
1. Heat kernel approach: good for differen-
tial operators which are positive;
2. Almost homogeneous calculus: good for
general hypoelliptic ΨHDO’s.
22
• Holomorphic Families of ΨHDO’s
Let Ω ⊂ C be open.
Definition. A family (qz)z∈Ω of symbols is
holomorphic when:
- The order mz is an analytic function of z;
- (qz) is a hol. family of smooth functions;
- The bounds of q ∼∑j≥0 qz,mz−j are locally
uniform in z.
Definition.A family (Qz) of ΨHDO’s is holo-
morphic if it is locally of the form
Qz = qz(x,−iX) +Rz,
where (qz) is a holomorphic family of symbols
and (Rz) is a holomorphic family of smooth-
ing operators.
23
• Heat Kernel Approach to Complex Powers
Let P be a differential operator of even Heisen-
berg order m such that:
- P is (semi-)positive, i.e. 〈Pu, u〉 ≥ 0;
- P has an invertible principal symbol.
Then for any s ∈ C we can define P s by stan-
dard L2-functional calculus.
Theorem (RP ’04).The above complex pow-
ers form a holomorphic family of ΨHDO’s.
24
The proof is quite easy (and again new). It
relies on combining the Mellin formula,
P−s = Γ(s)−1∫ ∞0
tse−tPdt
t, <s > 0,
together with an extension of the pseudodif-
ferential representation of the heat kernel of
P due to Beals-Greiner-Stanton (JDG ‘84).
This result applies to all our semi-positive ex-
amples (Kohn Laplacian, contact Laplacian)
and it has several interesting consequences:
25
- Fill a gap in the proof of Julg-Kasparov of
BC conjecture for SU(n,1) (complex powers
of the contact Laplacian).
- Heat kernel asymptotics for hypoellip-
tic differential operators, extending results
of Beals-Greiner-Stanton for (positive) sub-
laplacians.
- Construction of weighted Sobolev spaces
W sH(M), s ∈ R, interpolating Folland-Stein
spaces at positive integers and providing us
with sharp regularity estimates for ΨHDO’s.
26
• ΨHDO Representation of the Resolvent
We can give a pseudodifferential representa-
tion of the resolvent using an “almost homo-
geneous” ΨHDO calculus with parameter.
In this context the parameter set is a pseu-
docone, i.e a subset Λ ⊂ C\0 of the form
Λ = Θ ∪B with Θ conical and B bounded.
27
Let Λ be an open pseudocone and let p ∈ Z.
Definition. Holp(Λ) is the space of functions
f ∈ Hol(Λ) so that for any closed pseudocone
Λ′ ⊂ Λ we have
|f(λ)| ≤ CΛ′(1 + |λ|)p, λ ∈ Λ′.
Let w ∈ N∗ and let U be a Heisenberg chart
with H-frame X0, . . . , Xd. We define para-
metric ΨHDO’s on U as follows.
28
• Parametric symbols of order m:
Symbols q(λ) ∈ C∞(U ×Rd+1)⊗Holp(Λ) with
an asymptotic exapansion,
q(λ) ∼∑j≥0
q(λ),m−j,
where:
- q(λ),l ∈ C∞(U × (Rd+1 \ 0)) ⊗ Holp(Λ) is
almost homogeneous of degree l, i.e. for any
t > 0 we have
q(twλ),l(x, t.ξ)−tlq(λ),l(x, ξ) ∈ S(Rd+1)⊗Holp(Λ);
- the sign ∼ means a symbol asymptotics
whose bounds grow as O(|λ|p) with λ.
29
• Parametric ΨHDO’s of order m:
The class Ψm,pH,w(U) consists of families Q(λ)
of the form,
Q(λ) = q(λ)(x,−iX) +R(λ),
where q(λ) is a parametric symbol of order m
and R(λ) belongs to Ψ−∞(U)⊗Holp(Λ).
Proposition. 1) If Pj ∈ Ψmj,pjH,w (U), j = 1,2,
then P1P2 ∈ Ψm1+m2,p1+p2H,w (U).
2) The class Ψm,pH,w(U) is invariant by Heisen-
berg diffeomorphisms.
Thanks to the 2nd part we can define para-
metric ΨHDO’s on any Heisenberg manifold.
30
Let P be a ΨHDO of order m and let L be
ray in C \ 0.
Definition. L is a principal cut for P if near
any point x ∈M there are:
- a Heisenberg chart U ,
- an open pseudocone Λ containing L,
such that on U ×Rd+1×Λ the principal sym-
bol of P − λ is invertible.
Remark. The definition depends only on the
principal symbol of P and implies the invert-
ibility of the principal symbol P .
31
Definition. Θ(P ) is the union set of all the
principal cuts of P .
The principal set Θ(P ) is a cone which is
open when M is compact.
Assume Θ(P ) is not empty. Then:
Proposition (RP ’04). 1) The spectrum of
P consists in a discrete and unbounded set
of eigenvalues.
2) The intersection of SpP with any closed
cone Θ ⊂ Θ(P ) is finite.
32
Let Θ(P ) be the cone obtained from Θ(P ) by
removing from it all its rays that are through
an eigenvalue of P and set
Λ(P ) = Θ(P ) ∪ [D(0, R0) \ 0].
where RP = dist(0,SpP \ 0).
Theorem. 1) Λ(P ) is an open pseudocone.
2) (P − λ)−1 belongs to Ψ−m,−1H,m (M ; Λ(P )).
33
Cayley Hamilton Decomposition:
Assume Θ(P ) nonempty. The characteristic
suspace and projector associated to an eigen-
value λ ∈ SpP are
Eλ(P ) = ∪k≥0 ker(P − λ)k,
Πλ(P ) =∫|µ−λ|=r
(P − µ)−1dµ,
where r is small enough to isolate λ from the
rest of the spectrum.
Proposition (RP ‘04). 1) Πλ(P ) is a smooth-
ing operator which projects onto Eλ(P ) along
Eλ(P∗)⊥.
2) Eλ(P ) is a finite dimensional subspace
of C∞(M).
34
We extend the previous definitions to the
eigenvalue λ = ∞ by letting
Π∞(P ) = limR→∞
∫|µ−λ|=R
µ−1P−1(P − µ)−1dµ,
E∞(P ) = imΠ∞(P ).
Theorem (RP ‘04). For any s ∈ R we have
L2(M) = uλ∈Sp∪∞Eλ(P ),∑λ∈Sp∪∞
Πλ(P ) = 1.
This follows from a general result about the
Cayley-Hamilton decomposition of compact
operators and closed operators with compact
resolvent on Hilbert spaces (RP’ 04).
35
• Partial Inverse:
Consider the characteristic space,
EC\0(P ) = uλ∈(SpT∪∞)\0Eλ(P ).
Definition. The partial of P , denoted P−1,
is the bounded operator on L2(M) that van-
ishes on E0(P ) and inverts P on EC\0(P ).
Theorem (RP ‘04). 1) The partial inverse
P−1 is a ΨHDO of order −m.
2) We have
PP−1 = P−1P = 1−Π0(P ),
(P k)−1 = (P−1)k, k = 1,2, . . . .
In particular, P−1 is a parametrix for P .
36
• Complex Powers:
Let Lθ = argλ = θ be a ray contained in
Λ(P ). Then we can define a bounded oper-
ator on L2(M) by letting
P sθ =1
2iπ
∫Γθλs(P − λ)−1dλ, <s < 0.
Proposition. We have
Ps1+s2θ = P
s1θ P
s2θ ,
P−kθ ) = P−k, k = 1,2, . . . ,
where P−k denotes the partial inverse of P k.
37
Proposition (RP ‘04). The family (P sθ )<s<0
is a holomorphic family of ΨHDO’s so that
ordP sθ = s.ordP .
Thanks to this for <s ≥ 0 we can directly
defined P sθ as a ΨHDO by letting
P sθ = P kP s−kθ ,
where k is any integer > <s. Then we get:
Theorem (RP ‘04). The family (P sθ )s∈C is
an analytic 1-parameter group of ΨHDO’s so
that ordP sθ = s.ordP and
P1θ = P, P−1
θ = P−1, P0θ = 1−Π0(P ).
38
• Consequences of this Approach:
- For θ = π yields uniform boundedness re-
sults along vertical stripes a ≤ <zb in terms
of W sH-Sobolev spaces and the aperture of
Λ(P ) around Lπ.
- Existence and unicity result for the heat
equation,∂tu(x, t) + Pxu(x, t) = v(x, t),u(x,0) = u0.
associated to W sH-data (v, u0).
39
IV. Noncommutative Residue Trace
Let (Md+1, H) be a compact Heisenberg man-
ifold.
We can construct a noncommutative residue
trace on ΨHDO’s of integer order which is a
complete analogue of the celebrated Wodzicki-
Guillemin noncommutative residue trace.
40
• Logarithmic Singularity
Proposition (RP ‘01).Let P ∈ ΨmH(M), m ∈
Z. Then:
1) Near the diagonal the kernel kP (x, y) of P
has a behavior near the diagonal of the form
kP (x, y) =−1∑
j=−(m+d+2)
aj(x, ψx(y))
− cP (x) log ‖ψx(y)‖+ O(1),
where aj(x, λ.z) = λjaj(x, z).
2) The coefficient cP (x) makes sense globally
as a density on M .
41
• Analytic Extension of the Trace:
If ordP < −(d + 2) then P is traceable and
we have
TrP =∫MkP (x, x).
Using an analytic continuation of the map
P → kP (x, x) we get:
Theorem (RP ’01). 1) The trace has a unique
analytic continuation P → TRP to ΨC\ZH (M).
2) TR[P1, P2] = 0 when ordP1 + ordP2 6∈ Z.
42
Let P be a ΨHDO of integer order m. Then
TR has essentially a pole at P :
Theorem (RP ’01). Let (Pz) be a holomor-
phic family of ΨHDO’s such that:
- P0 = P ;
- ordPz = z +m.
Then the map z → TRPz has a simple pole
at z = 0 in such way that
resz=0 TRPz = −∫McP (x).
43
Definition. The noncommutative residue of
P ∈ ΨZH(M) is
ResP =∫McP (x).
From the construction of the functional Res
we obtain:
Proposition (RP ’01, ’04). 1) Res is a local
functional, i.e. is given by integration of a
density.
2) We have
ordQ.ResP = resz=0 TRPQ−zθ
for any positive order ΨHDO Q with principal
cut Lθ .
3) Res is trace, i.e. Res[P1, P2] = 0.
44
On the other hand, a famous result of Wodz-
icki (‘84) and Guillemin (‘93) is:
Theorem. If M is connected then any trace
on Ψ(M) is proportional to the Wodzicki-
Guillemin noncommutative residue Res.
In the Heisenberg setting we get:
Theorem (RP ’04). If M is connected then
any trace on ΨZH(M) is proportional to the
(Heisenberg) noncommutative residue Res.
45
• Some Consequences of this Construction:
- Short proof of a result of Hirachi (‘04)
on the logarithmic singularity of the Szego
kernel.
- Zeta functions and spectral asymmetry
of hypoelliptic ΨDO’s and connection with a
conjecture of Fefferman-Hirachi.
- Heat kernel asymptotics for general hy-
poelliptic ΨHDO’s (via inverse Mellin trans-
form).
- Conformal variations of spectral invari-
ants associated to the conformal powers of
the ∆b-sublaplacian on a CR manifold (re-
cently constructed by Gover-Graham (‘03)).
- Area of 3-dimensional CR manifold.
46
• Quantized Calculus (Connes)
Let H be a Hilbert space. Then the following
table of equivalences holds.
Classical Infinitesimal Quantized CalculusComplex Variable Operator on H
Real Variable Selfadjoint OperatorInfinitesimal Variable Compact OperatorInfinites. of order α Compact Operator s.t.
µk(T ) = O(k−α)Integral
∫f(x)dx Dixmier Trace −
∫T
47
Here:
- µk(T ) is the (k+ 1)’th characteristic value
of the compact operator T , that is
µk(T ) = (k+ 1)’th eigenvalue of |T | =√T ∗T .
- The Dixmier trace is defined on infinitesimal
operators of order ≥ 1 and such that, for
T ≥ 0,
limN→∞
1
logN
∑k≤N
µk(T ) = L⇒ −∫T = L.
Thus −∫
vanishes on trace-class operators, hence
on infinitesimal operators of order > 1 (like
the integral).
48
Theorem (Connes ‘88). Let M be a com-
pact manifold and let P be a ΨDO on M of
negative order −m.
1) P is an infinitesimal operator of order mdimM .
2) If ordP = −dimM , then
−∫P =
1
dimMResP,
where Res denotes the Wodzicki-Guillemin
noncommutative residue trace.
49
Theorem (RP ‘01). Let (M,H) be a com-
pact Heisenberg manifold and let P be a ΨHDO
on M of negative order −m.
1) P is an infinitesimal operator of order mdimM+1.
2) If ordP = −(dimM + 1), then
−∫P =
1
dimM + 1ResP.
50
Consequence:
We can integrate any ΨHDO, even when it
is not an infinitesimal operator of order ≥ 1,
just by letting
−∫P =
1
dimM + 1ResP.
51
• Area of a CR manifold:
Let (M2n+1, θ) be a pseudohermitian mani-
fold and let ∆b be its sublaplacian.
For any f ∈ C∞(M) we have
−∫f∆−(n+1)
b =∫Mf(x)(dθ)n ∧ θ.
Thus ds =√
∆b recaptures the contact vol-
ume. This leads us to define
AreaθM = −∫ds2.
52
Theorem (RP ‘01). If dimM = 3 then
AreaθM = −∫ds2 =
∫MrM(x)dθ ∧ θ,
where rM denotes the Tanaka-Webster scalar
curvature of M .
Example. For S3 ⊂ C2 we get
Areaθ S3 =
π2
2√
2
Potential Application: Should yield a spec-
tral interpretation of the Einstein-Hilbert ac-
tion of a Lorentzian manifold with Fefferman
metric.
53
VI. Local Index Formula for Heisenberg Manifold
• The Local Index Formula in NCG
The local index formula ultimately holds in
a purely operator theoretic setting (Connes-
Moscovici ’95).
This allows us to recover the Atiyah-Singer
index formula for Dirac operators, as well as
many other examples.
This framework uses two main tools:
- Spectral triples;
- Cyclic cohomology.
54
• Spectral Triples:
A spectral triple is a triple (A,H, D) where:
- H is a Hilbert space together with a Z2-
grading γ : H+ ⊕H− → H− ⊕H+;
- A is an involutive unital algebra represented
in H and commuting with the Z2-grading γ;
- D is a selfadjoint unbounded operator on
H s.t. [D, a] is bounded ∀a ∈ A and of the
form,
D =
(0 D−
D+ 0
), D± : H∓ → H±.
55
Spectral Triple
The datum of D above defines an index map
indD : K0(A) → Z so that
indD[e] = ind eD+e,
for anyselfadjoint idempotent e ∈Mq(A).
- p-summability:
We say that D is p-summable when we have
µk(D−1) = O(k−1/p) as k → +∞,
56
- Dimension spectrum:
Let Ψ0D(A) be the algebra generated by the
Z2-grading γ and the δk(a)’s, a ∈ A, where δ
is the derivation δ(T ) = [|D|, T ] (assuming Ais contained in ∩k≥0 dom δk).
Definition.The dimension spectrum of (A,H, D)
is the union set of the singularities of all the
zeta functions ,
ζ(P ; z) = TrP |D|−z, P ∈ Ψ0D(A).
57
- Noncommutative residue trace:
Assuming p-summability and simple and dis-
crete dimension spectrum we define a non-
commutative residue trace on Ψ0D(A) by let-
ting:
−∫P = Resz=0 TrP |D|−z for P ∈ Ψ0
D(A).
This functional is local in the sense of non-
commutative geometry since it vanishes on
the elements of Ψ0D(A) that are traceable.
58
Theorem (Connes-Moscovici ‘95).Suppose
that (A,H, D) is p-summable and has a dis-
crete and simple dimension spectrum.
1) The following formulas define an even co-
cycle ϕCM = (ϕ2k) in the (b, B)-complex of
the algebra A.
- For k = 0,
ϕ0(a0) = finite part of Tr γa0e−tD
2as t→ 0+,
- For k 6= 0,
ϕ2k(a0, . . . , a2k) =
∑αck,α −
∫γPk,α|D|−2(|α|+k),
Pk,α = a0[D, a1][α1] . . . [D, a2k][α2k],
where the ck,α’s are universal rational con-
stants and the symbol T [j] denotes the j’th
iterated commutator with D2.
2) We have:
indD[e] = 〈[ϕCM], e〉 ∀e ∈ K0(A).
where 〈., .〉 is the pairing of cyclic cohomology
with K-theory.
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• Index Formula for Heisenberg manifolds:
Here we consider:
- A (compact) Heisenberg manifold (M,H);
- A Z2-graded bundle S = S+ ⊕ S− over M
with grading γ;
- An operator D ∈ Ψ1H(M,S) which anti-
commutes with γ and such that the complex
powers of D2 form a holomorphic family of
ΨHDO’s;
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Using the complex powers of D2 and the non-
comutative residue for the Heisenberg calcu-
lus we get:
Proposition (RP ‘01). The spectral triple,
(C∞(M), L2(M,S), D)
is (d+2)-summable and has a simple dimen-
sion spectrum contained in
k ∈ Z; k ≤ dimM + 1.
Therefore, the theorem of Connes and Moscovici
applies.
In this setting the index formula has a geo-
metric description as follows.
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Let E be a Hermitian vector bundle together
with a unitary connection ∇. Then:
a) Since here A = C∞(M) the CM cocycle is
actually a current CD whose pairing with the
K-theory class of E is given by
〈[CD], [E]〉 = 〈CD,ChF E〉,
where ChF E = Tr e−FE
is the total Chern
form of the curvature F E of E.
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b) Using ∇ we can twist D into D∇,E given
by the composition
Γ(S ⊗ E) 1E⊗∇→ Γ(S ⊗ T ∗M ⊗ E) πD⊗1E→ Γ(S ⊗ E),πD[(f0df1)⊗ σ] = f0[D, f1]σ.
This operator anticommutes with the grad-
ing of S ⊗ E and we have
indD[E] = indD+∇,E .
Therefore, we obtain:
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Theorem (RP ‘01). 1) There exists an even
de Rham current CD on M such that for any
Hermitian vector bundle E over M with uni-
tary connection ∇ we have
indD+∇,E = 〈CD,ChF E〉.
2) The components C2k, k = 0,2, . . ., of CDare given by the following formulas.
- For k = 0,
〈C0, f0〉 =
∫Mf0(x)StrS a0(D
2, x),
- For k 6= 0,
〈C2k, f0df1 ∧ . . . ∧ df2k〉 =
∑αck,α −
∫γPk,α|D|−2(|α|+k),
Pk,α = f0[D, f1][α1] . . . [D, f2k][α2k],
where −∫
denotes the noncommutative residue
for the Heisenberg calculus.
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• Geometric Operators
There is an obstruction to constructing sig-
nificant geometric operators for which the
previous index formula applies. Namely, we
cannot transplant into the CR and contact
settings the Dirac construction used by Aitiyah-
Singer, for it yields non-hypoelliptic opera-
tors. However, using a construction inspired
by that of the mixed transverse signature op-
erator of Connes-Moscovici (GAFA ‘95), we
can build hypoelliptic signature-type opera-
tors out of:
- the Kohn-Rossi complex (RP’ 01);
- Rumin’s contact complex (RP unpub-
lished).
The actual computation of the correspond-
ing currents is still in progress.
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