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.

59

• 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;

60

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.

61

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:

63

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.

64

• 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.

65