The Gauss-Bonnet theorem for the noncommutative 2-torus∗ Francesca Arici and Domenico Monaco July 19th, 2012

Contents 1 Pseudodierential operators 1.1

1

Pseudodierential operators on manifolds . . . . . . . . . . . . . . . . . .

2 The heat kernel proof of the Gauss-Bonnet theorem

3

4

2.1

The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

The asymptotics of the heat kernel

5

. . . . . . . . . . . . . . . . . . . . .

3 The noncommutative 2-torus

8

3.1

Basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Pseudodierential operators

3.3

Spectral theory and parametrices

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Understanding the computations

8 10 11

16

4.1

The zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

The Connes-Tretko way to compute

ζ(0)

. . . . . . . . . . . . . . . . .

16 18

1 Pseudodierential operators First of all we start by recalling some notation. Euclidean scalar product.

∗

Let

α = (α1 , . . . , αn ),

Our ambient space is with

αi ∈ N,

1

with the

be a multiindex.

In partial fulllment of the exam Introduction to noncommutative geometry. given by prof. Ludwik D¡browski during the period April  May 2012.

Rn

We

The lectures were

dene the following compact notation:

|α| := α1 + · · · + αn , α! = α1 ! · · · αn !, α1 α x = (x1 , . . . , xαnn ), Dxα = (−i)|α| dαx .

dαx = (∂1x )α1 · · · (∂nx )αn ,

Schwarz class S

Denition 1.1. functions

f

The n on R such that for all

is the vector space of all smooth complex-valued

α, β

there exist constants

Cα,β

such that

|xα Dβ f | ≤ Cα,β . This is equivalent to assuming that there exist estimates

|Dxβ f | ≤ Cn,β (1 + |x|)−n for all

(n, β).

Remark

1.1

.

F :S→S

Recall that the Fourier transform

mapping

f 7→ fˆ is

a homeo-

morphism and that it satises

Dξα F(f )(ξ) = (−1)|α| F(xα f ),

Denition 1.2.

A

ξ α F(f )(ξ) = F(Dxα f ).

linear partial dierential operator P

of order

d

(1)

is a polynomial

expression

X

P = P (x, D) =

aα (x)Dxα

|α|≤d where the coecients We dene the

aα (x)

symbol

belong to

∞

n

C (R ). X

σ(P ) := p(x, ξ) =

aα (x)ξ α

|α|≤d by formally replacing the dierential operator The

leading symbol

Dα

by the monomial

ξα.

is the highest order homogeneous part

σL (P ) := pd (x, ξ) =

X

aα (x)ξ α .

|α|=d By the properties of the Fourier transform (1) we can express the operator integral operator

1

P

as an

by

Z P f (x) =

e

ix·ξ

p(x, ξ)fˆ(ξ)dξ =

ZZ

ei(x−y)·ξ p(x, ξ)f (y)dydξ.

(2)

Therefore we can dene the (larger) class of pseudodierential operators by replacing the polynomial

p

1 Use the fact that

in (2) by a more general symbol.

Dxα f (x) = F −1 ξ α fˆ(ξ).

2

Denition 1.3. functions

Cα,β

We dene the set

Sd

of

symbols of order d

to be the set of smooth

p(x, ξ) with compact x-support and such that for all (α, β) there exist constants

so that

|Dxα Dξβ p(x, ξ)| ≤ Cα,β (1 + |ξ|)d−|β| . d Let p ∈ S . Cc∞ (Rn ) by

In analogy to Equation (2), we dene the associated operator

Z P f (x) = Ψd be order d. Let

(3)

e

ix·ξ

p(x, ξ)fˆ(ξ)dξ =

ZZ

P : S →

ei(x−y)·ξ p(x, ξ)f (y)dydξ.

(4)

pseudodierential operators of

the space of such operators. They are called

Observe that, while the order of a dierential operator is an integer, the order of a pseudodierential operator need not be an integer. To be able to extend our denition to manifolds we must consider the restriction of U ⊂ Rn be an open subset with compact closure and d let U ⊂ O where O has also compact closure. We consider the space S (U ) of symbols d with compact x-support in U and corresponding operators Ψ (U ). We dene the algebra our operators to open subsets. Let

of symbols and corresponding pseudodierential operators

S(U ) =

[

S d (U ),

Ψ(U ) =

[

Ψd (U )

d

d

and the ideal of smoothing symbols and the corresponding smoothing operators

S −∞ (U ) =

\

Ψ−∞ (U ) =

S d (U ),

p−q ∈S

−∞

Ψd (U ).

d

d

Denition 1.4.

\

We say that two symbols

p

and

q

are

equivalent,

p ∼ q,

and write

i

.

1.1 Pseudodierential operators on manifolds We want to extend the previous notions to manifolds. In order to do so we will study elliptic pseudodierential operators.

Denition 1.5.

Let

p ∈ S d (U ).

Let

U1 ⊂ U .

We say that

p

is

elliptic

on

U1

if one of

the following equivalent conditions is satised:

q ∈ S −d (U ) and a plateau function φ ∈ Cc∞ (U ) φ(qp − I) ∈ S −∞ (U ) and φ(pq − I) ∈ S −∞ (U ).

identically

1

near

U1

q ∈ S −d (U ) and a plateau function φ ∈ Cc∞ (U ) φ(qp − I) ∈ S −1 (U ) and φ(pq − I) ∈ S −1 (U ).

identically

1

near

U1

(a) There exists such that

(b) There exists such that

(c) There exists an open set U2 which contains |p(x, ξ)|−1 ≤ C1 (1 + |ξ|)−d for |ξ| ≥ C0 and x

3

U1 and ∈ U2 .

positive constants

Ci

so that

Remark

1.2

.

We observe that adding lower order terms does not alter the ellipticity.

Moreover if a polynomial

p

is the symbol of a dierential operator

i the leading symbol is invertible for

Denition 1.6. elliptic on Let

M

Let

U1 ⊂ U .

P,

then

p

is elliptic

ξ 6= 0.

An operator

P

is said to be

elliptic

on

U1

if its symbol is

U1 . be a smooth compact Riemannian manifold without boundary of dimension

m.

Denition 1.7.

∞ ∞ Let P : C (M ) → C (M ). If (O, h) is a coordinate chart and if φ, ψ ∈ ∞ ∞ Cc (O), the operator φP ψ on Cc (O) is called a localization of P . Via pushforward we ∞ dene an operator h∗ (φP ψ) on Cc (h(O)). We say that P is a d M and write P ∈ Ψd (M ) if

of order

pseudodierential operator

on

h∗ (φP ψ) ∈ Ψd (h(O))

for all

φ, ψ, O, h.

We analogously dene the space of symbols of order

d

on

M

and their equivalence

relation.

Denition 1.8.

We say

Let

S d = {p(x, ξ) ∈ C ∞ (T ∗ M ) : h∗ (φp) ∈ S d (hO) ∀O, h, φ}. \ S d (M ). p1 ∼ p2 if p1 − p2 ∈ S −∞ (M ) = d

We dene the space of square integrable functions using the

L2

inner product

Z hf, gi =

f (x)g(x)dvol(x). M

We set

kf k2 := hf, f i

and dene

L2 (M )

to be the completion of

Cc∞ (M )

in this norm.

2 The heat kernel proof of the Gauss-Bonnet theorem Let again

M

be a compact Riemannian manifold. Consider a smooth vector bundle d V → M . Let d > 0. We denote by Pell,s (M, V ) the set of symmetric (i.e. formally ∞ self-adjoint) elliptic partial dierential operators on C (V ). Moreover we denote by d,+ Pell,s (M, V ) the subset of operators such that σL (P )(x, ξ) is positive denite for all

ξ 6= 0. d Lemma 2.1. Let P ∈ Pell,s (M, V ). Then there exists a

discrete spectral resolution

P , i.e. there exist {φn , λn } with φn ∈ C (V ) so that P φn = λn φn . ∞

4

for

2.1 The heat equation Let

d,+ P ∈ Pell,s (M, V ).

The heat equation associated to

P

is the system of equations

( (∂t + P )h(x, t) = 0, lim h(x, t) = f (x),

(5)

t→0

for

t > 0.

{φn , λn } f ∈ C ∞ (V )

Let

decompose

be a discrete spectral solution for

P

as in Lemma 2.1. We may

as

f=

X

cn = hf, φn i.

cn φn ,

n Dene

φ∗n ∈ C ∞ (V ∗ )

φ∗n (f ) = hf, φn i. We set X k(t, x, y) := e−tλn φn (x) ⊗ φ∗n (y) ∈ Vx ⊗ Vy∗ = Hom(Vy , Vx ) by

n

e

−tP

f (x, t) :=

X

e

−tλn

Z cn φn (x) =

k(t, x, y)f (y)dvol(y). M

n

Lemma 2.2. he function h(x, t) = e−tP f (x) is a solution to (∂t + P )h(x, t) = 0 and h(x, t) converges uniformly as t & 0 to f (x) in C k for all k .

2.2 The asymptotics of the heat kernel d,+ Theorem 2.3. Let P ∈ Pell,s (M, V ). Then

(a) If t > 0, then the operator e−tP is an innitely smoothing operator which is of trace class on L2 . (b) As t & 0 there is a asymptotic expansion TrL2 (e−tP ) ∼

X

an (P )t

n−m d

.

(6)

n

(c) The heat trace asymptotics coecient an (P ) vanishes for n odd. If n is even there exist local invariants en (x, P ) such that Z an (P ) =

tr(en (x, P ))dvol(x), M

where tr denotes the ber trace. Recall the denition of the de Rham complex and its cohomology.

5

Denition 2.1.

The

k -th de Rham cohomology group

of a manifold

M

is given by

k HdR (M ) = {ω ∈ Ωk : dω = 0}/{dθ : θ ∈ Ωk−1 }. The dimension of

k HdR (M ) as a vector space is denoted by β k

and is called the

k -th Betti

number.

Denition 2.2.

The Laplacian on

Ωk (M )

is given by

4k = δ k+1 dk + dk−1 δ k . Elements of the kernel of the Laplacian on forms are called

harmonic k-forms.

Every

harmonic form is closed and this gives a linear map

k h : ker 4k → HdR (M )

ω 7→ [ω].

This leads to the following formulation of Hodge's theorem.

Theorem 2.4

isomorphism

Corollary 2.5

(Hodge)

. Let M be a compact Riemannian manifold. Then h yields an k ker 4k ' HdR (M ).

. Moreover, if M is oriented, the Hodge ∗ operator

(Poincarè duality)

yields the isomorphism

m−k k HdR (M ) ' HdR (M ).

M is X X χ(M ) := (−1)i β i = (−1)i dim ker 4i .

As a consequence, the Euler characteristic of

i

i

The proof of Hodge's theorem for forms relies on the denition of the corresponding heat kernel.

Theorem 2.6

. The Laplacian on forms satises

(Weitzenböck formula)

4 = ∇∗ ∇ + K.

Theorem 2.7

(Bochner's formula)

(7)

. The Laplacian on 1-forms satises 41 = ∇∗ ∇ + K.

(8)

Theorem 2.8 (Gauss-Bonnet). Let M a closed oriented surface with Gaussian curvature K and area element dA. Then

1 χ(M ) = 2π

6

Z KdA. M

(9)

The proof of the Gauss Bonnet theorem relies on a crucial observation by McKean + k k and Singer [4]. Let λ ∈ R . Denote by Eλ the λ-eigenspaces for 4 . The dierential dk k k+1 2 intertwines 4 and 4 . Indeed by denition of 4 and by the property that d = 0 we have

4k+1 ◦ dk = δk+1 ◦ dk+1 ◦ dk + dk ◦ δk+1 ◦ dk = dk ◦ (4k − dk−1 ◦ δk ) = dk ◦ 4k . Let

ω ∈ Eλk .

Then one has

4k+1 ◦ dk ω = dk ◦ 4k ω = dk (λω) = λdω, which means that if

ω ∈ Eλk ,

dω ∈ Eλk+1 .

then

Hence we have a sequence

d

k 0 → Eλ0 → · · · → Eλk −→ Eλk+1 → · · · → Eλm → 0.

k Moreover the sequence is exact. Indeed let ω ∈ Eλ k λω = 4 ω = (dδ + δd)ω = dδω which implies that ω =

∩ ker(dk ). Then by denition d(λ−1 δω) ∈ Ran(dk−1 ).

This implies that

X

(−1)k dim(Eλk ) = 0

for

λ 6= 0.

(10)

k Let us compute

X

k

(−1)k Tr(e−t4 ) :=

X

k

(−1)k

X

k

e−λi t =

X

i

k

(−1)k

X

dim(Eλki )e−λi t .

(11)

i

k

λi dierent. By (10) the only term that counts is the one with λ = 0, therefore X X X k (−1)k Tr(e−t4 ) = (−1)k dim(E0k ) = (−1)k dim ker 4k = χ(M ). (12)

with all

k

k

k

This implies that

X

k

(−1)k Tr(e−t4 )

k is independent of

t.

Therefore its long-time behaviour is the same as its short-time

behaviour. In particular, if we apply the heat kernel asymptotics (Theorem 2.3) we get

χ(M ) =

X

(−1)k dim ker 4k =

t→0

where

ekn

k

(−1)k Tr(e−t4 ) =

k

k

= lim

X

X k

k

(−1)



X Z n

tr ekn (t, x, x)dvol(x)

M

are the coecients of the asymptotic expansion of

pendent of

t,

t(n−m)/2 k

Tr(e−t4 ).

Since

χ

is inde-

only the constant term on the right-hand side can be nonzero, i.e.

X k

(−1)k

( 0 tr(ekn (x))dvol(x) = χ(M ) M

Z

7

if if

n 6= m, n = m,

m

even.

(13)

In particular for a two-dimensional Riemannian manifold the only term that counts is the one involving

e2 (x).

Therefore we have

Z


tr e02 (x) − tr e12 (x) + tr e22 (x) dvol(x).

χ(M ) =

(14)

M 1 1 K − K where K denotes the scalar curvature 2iπ 6 and K is curvature operator appearing in the Weitzenböck formula. By Hodge duality tr(e02 ) = tr(e22 ). In particular one can compute




e2 (x) =

One can check that

tr(e02 )

=

tr(e22 )

1 K = , 4π 6

tr(e12 )

1 = 4π



 2 − K , 3

from which the claim follows.

3 The noncommutative 2-torus 3.1 Basic denitions ∗ In this section, we review the denition and basic properties of the C -algebra of functions over the noncommutative 2-torus (also known as the irrational rotation algebra). Fix an irrational number θ (actually it suces to consider θ ∈ [0, 1/2] ∩ (R \ Q)). The T2θ is dened as the noncommutative topological space whose algebra of continuous functions is generated by two elements U, V satisfying

noncommutative 2-torus

U ∗ = U −1 ,

V ∗ = V −1

(i.e.

VU =e

U

2πiθ

X

V

are unitaries),

U V.

∗ In other words, every element in the C -algebra

a=

and

C(T2θ ) ≡ Aθ

anm U n V m ,

is a formal series

anm ∈ C.

(15)

(n,m)∈Z2 ∗ We introduce a C -dynamical system on T2 = R2 /(2πZ)2 determined by

Aθ

given by the action of the usual torus

αx (U n V m ) = eix·(n,m) U n V m , This allows us to dene the sub-algebra

C ∞ (T2θ ) ≡ A∞ θ

x ∈ R2 .

(16)

consisting of those

a ∈ Aθ

such

that the map

x ∈ R2 7→ αx (a) ∈ Aθ is smooth. Explicitly, if

a

is the series (15) then its Fourier coecients

rapidly at innity, i.e.

sup |n|p |m|q |anm | < +∞ (n,m)∈Z2

8

for all

p, q ∈ N.

anm

must decay

The derivations which constitute the innitesimal generators of the action

α

are deter-

mined by

δ1 (U ) = U, δ2 (U ) = 0,

δ1 (V ) = 0, δ2 (V ) = V. C(T2 ) = A0 , these U = eix1 , V = eix2 are

For functions in the commutative algebra operators functions As

θ

δ1 = −i∂x1 , δ2 = −i∂x2 2 on T ).

(if

is irrational, there is a unique trace on

Aθ

would correspond to the the standard coordinate

(see e.g. [3, Proposition 12.11]) given

by

τ (a) = a00

for

X

a=

anm U n V m .

(n,m)∈Z2 We can thus construct the GNS Hilbert space completion of

Aθ

H0

τ:

associated to

explicitly, this is the

with respect to the inner product

ha, bi = τ (b∗ a),

a, b ∈ Aθ .

We introduce the following two (unbounded) operators on

(17)

H0 :

∂ ∗ = δ1 − iδ2 .

∂ = δ1 + iδ2 ,

∂ ∗ is the adjoint of ∂ with respect to the inner product (17). We also dene (1,0) 2 ∞ the space H of noncommutative (1,0)-forms on Tθ as the unitary bi-module over Aθ ∞ generated by elements of the form a∂b with a, b ∈ Aθ , and completed with respect to

Clearly

the inner product

ha1 ∂b1 , a2 ∂b2 i(1,0) = τ (a∗2 a1 (∂b1 )(∂b2 )∗ ) . The operator

∂

can now be seen as an (unbounded) operator on

H0

with values in

H(1,0) .

Moreover, setting

H = H0 ⊕ H

(1,0)

 ,

one can show (see e.g. [1, Lemma 2.2]) that Laplace operator associated to

D

D=

0 ∂∗ ∂ 0

(Aθ , H, D)



is an even spectral triple. The

is

4 = ∂ ∗ ∂ = δ12 + δ22 . This Laplacian is clearly self-adjoint and positive semidenite. Notice that ker 4 = ker ∂ . ∂a = 0 then 4a = ∂ ∗ ∂a = 0, so that ker ∂ ⊆ ker 4. Conversely, if

Indeed, clearly if

4a = 0

then

0 = ha, ∂ ∗ ∂ai = h∂a, ∂ai = k∂ak2 so that

∂a = 0,

and

ker 4 ⊆ ker ∂ .

We now compute

ker ∂

as follows:

0 = ∂(U n V m ) = δ1 (U n V m ) + iδ2 (U n V m ) = nU n−1 V m + imU n V m−1 ⇔ n = 0 = m and hence

dim ker ∂ = dim ker 4 = 1.

9

(18)

3.2 Pseudodierential operators In this section, we establish the notations and recall a few useful results from the theory of pseudodierential operators on the noncommutative 2-torus. We start from dierential operators of order of

δ1 , δ2

which are at most of degree

X

P =

d,

namely those polynomial expressions

d:

aI δ I ,

aI ∈ A∞ θ ,

δ I := δ1i1 δ2i2 .

I=(i1 ,i2 )∈N2 |I|≡i1 +i2 ≤d

ˆ2 R

R2 , with 2 coordinates (ξ1 , ξ2 ); we will denote by ∂i the operator ∂/∂ξi , and for J = (j1 , j2 ) ∈ N j j 1 2 J ∞ ˆ2 we set ∂ := ∂1 ∂2 . An algebra-valued smooth function ρ = ρ(ξ1 , ξ2 ) ∈ C (R ; A∞ θ ) is called a symbol of order d ∈ Z if the following conditions hold: We extend this denition in the following way. Let

•

for all

I, J ∈ N2

I J

δ ∂ ρ(ξ) ≤ Cρ,I,J (1 + |ξ|)d−|J|

for some constant

•

denote the group dual to

Cρ,I,J ;

there exist a function

ˆ 2 \ {(0, 0)}; A∞ ) ρd ∈ C ∞ (R θ

such that

lim µ−d ρ(µξ) = ρd (ξ).

µ→∞ We denote by

S d the space of symbols of order d; the union2 S =

To every symbol

ρ∈S

we may associate an operator

formula

Pρ

[

S d forms an algebra.

d∈Z acting on

A∞ θ

by the general

ZZ

dx dξ −ix·ξ e ρ(ξ)αx (a). 2 ˆ 2 (2π) R2 × R X simplies: if a = anm U n V m

Pρ (a) = In view of (16), this formula

(19) then

(n,m)∈Z2

Pρ (a) =

X ZZ (n,m)∈Z2

ˆ2 R2 × R

dx dξ −ix·ξ e ρ(ξ)anm αx (U n V m ) = 2 (2π)

X ZZ

dx dξ −ix·ξ e ρ(ξ)eix·(n,m) anm U n V m = 2 (2π) 2 2 ˆ R ×R (n,m)∈Z2 Z  Z X dx −ix·[ξ−(n,m)] dξ ρ(ξ) = e anm U n V m = 2 (2π) 2 2 ˆ R (n,m)∈Z2 R   Z X = dξ ρ(ξ)δ(ξ − (n, m)) anm U n V m = =

(n,m)∈Z2

=

X

ˆ2 R

ρ(n, m)anm U n V m .

(n,m)∈Z2 2 We will not need symbols of non-integer order in the following.

10

pseudodierential operators

All operators which arise from such symbols are called . ∗ The class of pseudodierential operators forms a -algebra, namely it is closed under taking adjoints and composition. The order of a pseudodierential operator is the order of its symbol. One can reconstruct the symbol back from a pseudodierential operator, but this symbol is unique only up to the equivalence

ρ ∼ ρ0

if and only if

ρ − ρ0

is a symbol of order

Notice in particular that, given a sequence of symbols relation

ρ∼

∞ X

d

for all

ρj ∈ Sdj

d ∈ Z.

with

dj & −∞,

the

ρj

j=0

d ∈ Z there exists an M (d) such that for all M ≥ M (d) the dierence ρ − ρ0 − . . . − ρM is a symbol of order d; no convergence is required on the series in the right-hand side of the above equation. We will denote by σ(P ) the symbol associated to a pseudodierential operator P (dened only modulo the equivalence relation ∼). holds if for all

Equation (19) can be rewritten as

ZZ P (a) = ˆ2 R2 × R if

X

a=

dx dξ −ix·ξ e σ(P )(ξ)αx (a) = (2π)2

X

σ(P )(n, m)anm U n V m ,

(20)

(n,m)∈Z2

anm U n V m .

(n,m)∈Z2

d is called ∞ of its symbol is invertible inside the algebra Aθ for large

As a last denition, recall that a pseudodierential operator

elliptic |ξ|,

if the value

σ(P )(ξ)

P

of order

and its inverse satises



σ(P )(ξ)−1 ≤ C1 (1 + |ξ|)−d for some constants operator

4,

C0 , C1

for

P . An example σ(4)(ξ) = |ξ|2 .

depending only on

whose invertible symbol is

|ξ| ≥ C0 is given by the Laplace

3.3 Spectral theory and parametrices For notational convenience, in the following we will concentrate on pseudodierential

d which are elliptic, self-adjoint, and positive semidenite (as is the case of P = 4 for d = 2). We let R denote the resolvent set of P : our goal is to study −1 the resolvent operator (P − λ) for λ ∈ R. Consequently, we will need a wider class d of symbols, depending also on the parameter λ. Thus, we will call SN (R) the space of ∞ 2 ˆ and λ ∈ R, such that: functions η = η(ξ, λ) ∈ Aθ , for ξ ∈ R ˆ 2 , the function λ 7→ η(ξ, λ) is holomorphic; • for all xed ξ ∈ R operators of order

•

for xed

λ ∈ R,

the function

ξ 7→ η(ξ, λ)

is

C ∞,

and for all

I, J ∈ N2

one has

I J l

δ ∂ ∂λ η(ξ, λ) ≤ Cη,I,J,l (1 + |ξ| + |λ|1/d )N −|J|−dl for some constant

Cη,I,J,l .

11

and

l∈N

Thus, for xed

λ∈R

η(ξ, λ) is a symbol of order N , but a stronger decay |λ|. Moreover, λ is weighted with degree d  we will have more

the function

rate is required for big

to say about this in a moment. To every such symbol

η,

we may associate an operator on

ZZ Rη (λ)(a) =

ˆ2 R2 × R

A∞ θ

by setting

dx dξ −ix·ξ e η(ξ, λ)αx (a). (2π)2

Conversely, to associate a unique symbol to every such operator we need to quotient

σ ). Due to the λ, more care has to be taken

by an equivalence relation (the symbol map will be denoted again by requirement of decay also of the derivatives with respect to

in keeping track of the orders involved (see [2] for more details). Hence we dene

η ∼N η 0

if and only if

d η − η 0 ∈ S−N (R)

and in particular

∞ X

η ∼N

ηj

if and only if

d η − η0 − · · · − ηM ∈ S−N (R)

for

M ≥ M (N ).

j=0

η ∼ η 0 if η ∼N η 0 for all N ∈ Z. d η ∈ SN (R) is called homogeneous of

We will say that A symbol

order

N

in

(ξ, λ)

if

η(νξ, ν d λ) = ν N η(ξ, λ). ν in front of λ: roughly speaking, λ is considered as a symbol d. We also say that an operator R = R(λ) of order N is polyhomogeneous if for j ∈ Z we may expand its symbol as

Please notice the power of of order all

σ(R) ∼N −j r0 + r1 + · · · + rj where each

d r l ∈ SN −l

is homogeneous of order

N −l

in

(ξ, λ).

We are now ready to give the main result that we will need in the following.

Theorem 3.1. Let P be an elliptic, self-adjoint, and positive semidenite pseudodif-

ferential operator of order d. Then for every N ∈ Z there exists a polyhomogeneous operator RN = RN (λ) of order −d, called a parametrix, such that (P − λ)RN (λ) − I ∼N 0,

RN (λ)(P − λ) − I ∼N 0.

Moreover, if σ(P ) = ρ0 + · · · + ρd as a sum of homogeneous terms (i.e. ρk ∈ S k /S k−1 ) then the homogeneous summands in σ(RN ) are found recursively by setting r0 = (ρd (ξ) − λ)−1   rl = −  

X |I|+d+n−k=l, n
12

and

(21a)

 ∂ I r n δ I ρk   r0 .  I!

(21b)

Proof.

The main idea of the proof is to construct the symbol of

its dening relation

RN (λ)(P − λ) ∼N I .

RN (λ) recursively using

The formula for the symbol of the product of

two operators is known, see [2, Lemma 1.2.3]:

X ∂ I ηδ I ρ I! 2

σ(QP ) ∼

I∈N where

ρ, η

are the symbols of

P, Q, respectively, and where if I = (i1 , i2 ) then I! = i1 !i2 !.

In our case, we obtain

σ(RN (λ)(P − λ)) ∼N where

ρek

X ∂ I rn δ I ρek I! I,n,k

are the homoegenous pieces in the symbol of

( ρk ρek = ρd − λ (here we use the fact that

λ

if if

P − λ,

(22)

namely

k = 0, 1, . . . , d − 1, k=d

is considered a symbol of order

d).

We know that the

expression on the right-hand side of (22) will be a sum of homogenous pieces of order

−l, l ≥ 0,

and that the left-hand side will also have to be equivalent to

which is homogenous of order homogeneous of order

0;

0.

σ(I) = 1,

So we rst restrict our attention to the term which is

this will give the leading (or principal) symbol

σL .

The latter

is multiplicative on operators, see [2, Lemma 1.3.4]: hence we get

1 = σ(I) = σL (I) ∼N σL (RN (λ)(P − λ)) = σL (RN (λ))σL (P − λ) = r0 (ρd − λ) and hence

r0 ∼N (ρd − λ)−1 which is Equation (21a) up to neglecting lower order terms. We now dene recursively the homogenous pieces of order to know all

rn 's

for

n = 0, 1, . . . , l − 1.

−l, l > 0,

so we assume

We have to take into account the degree of

homogeneity of all the terms in the sum appearing in (22). Now:

• rn is homogeneous −d − n − |I|; • ρk

of order

δ

and hence

∂ I rn

is homogeneous of order

I and hence also δ ρk is homogenous of order k (be∞ is a derivation in the algebra Aθ and hence does not involve derivatives

is homogeneous of order

cause

−d − n,

with respect to

k,

ξ ).

We conclude that the piece of the sum which is homogenous of degree

X −d−n−|I|+k=−l

∂ I rn δ I ρek = I!

13

X |I|+d+n−k=l

∂ I rn δ I ρek . I!

−l

is

k ≤ d and |I| ≥ 0, so if |I| + d + n − k = l then n ≤ l. We therefore isolate the terms with n = l in the above sum, which are the ones involving rl that we are trying to compute. If n = l , then k = |I| + d ≥ d; but we know that k ≤ d, so that we must have k = d and |I| = 0, or I = (0, 0). As 1 = σ(I) has no homogeneous terms

We observe that

of negative order, we conclude that

X

rl (ρd − λ) +

|I|+d+n−k=l, n
ρek

with

ρk

∂ I r n δ I ρk ∼N 0 I!

in the above sum because

n
implies

k < d)

which,

together with the already proven Equation (21a), gives exactly (21b).

P =4

We will now apply these considerations to the case order

d = 2),

(which is an operator of

and compute explicitly the rst terms in (21a), (21b) up to

adopt the same notations of [1], we let

σ(4)(ξ) = a0 (ξ) + a1 (ξ) + a2 (ξ) and

σ(R2 (λ))(ξ) ∼2 b0 (ξ, λ) + b1 (ξ, λ) + b2 (ξ, λ). Clearly (21a) gives

b0 (ξ, λ) = (a2 (ξ) − λ)−1 (compare [1, Equation (26)]). We now compute



 I

X

 b1 = − 

|I|+2+n−k=1, n<1 (because

n<1

implies

n = 0).



I

I

I



X ∂ bn δ ak ∂ bn δ ak   b0  b0 = −  I! I! |I|+1=k

We recall that

k = 0, 1, 2,

•

when

k = 0,

we have

|I| = −1

•

when

k = 1,

we have

|I| = 0,

or

I = (0, 0);

•

when

k = 2,

we have

|I| = 1,

or

I = (1, 0)

and hence

which is impossible;

and

I = (0, 1).

In conclusion

k=1

k=2

z }| { z }| { b1 = −( b0 a1 b0 + ∂1 b0 δ1 a2 b0 + ∂2 b0 δ2 a2 b0 ) = | {z } | {z } | {z } I=(0,0)

I=(1,0)

= −(b0 a1 b0 + ∂i b0 δi a2 b0 )

14

I=(0,1)

l = 2.

To

where summation over the repeated index

i ∈ {1, 2} is understood (compare [1, Equation

(27)]). Lastly we compute







 X ∂ I bn δ I ak  ∂ I bn δ I ak   b0 .  b0 = −    I! I!

X

 b2 = − 



|I|+2+n−k=2, n<2

|I|+n=k n=0,1

We argue as before:

•

when

   •

we obtain

|I| = k

and hence

if

k=0

then

I = (0, 0);

if

k=1

then

I = (1, 0)

if

k=2

then

I = (2, 0), I = (1, 1)

when

  

n=0

n=1

we obtain

and

I = (0, 1);

|I| = k − 1

if

k=0

then

|I| = −1

if

k=1

then

I = (0, 0);

if

k=2

then

I = (1, 0)

and

I = (0, 2);

and hence

which is impossible;

and

I = (0, 1).

Consequently

b2 =

k=1  z k=0 }| { z }| { − b0 a0 b0 + ∂1 b0 δ1 a1 b0 + ∂2 b0 δ2 a1 b0 + | {z } | {z } | {z } I=(0,0)

I=(1,0)

I=(0,1)

          

n=0 z }| {    1 1  + ∂12 b0 δ12 a2 b0 + ∂1 ∂2 b0 δ1 δ2 a2 b0 + ∂22 b0 δ22 a2 b0 +   | {z } 2 2   {z } | {z } |  I=(1,1) I=(2,0) I=(0,2)  k=2 k=1 }| {  z }| { z + b1 a1 b0 + ∂1 b1 δ1 a2 b0 + ∂2 b1 δ2 a2 b0 = n = 1 | {z } | {z } | {z }  k=2

I=(0,0)

I=(1,0)

I=(0,1)



 1 = − b0 a0 b0 + b1 a1 b0 + ∂i b0 δi a1 b0 + ∂i b1 δi a2 b0 + ∂i ∂j b0 δi δj a2 b0 , 2 where summation over the repeated indices Equation (28)]).

15

i, j ∈ {1, 2}

is understood (compare [1,

4 Understanding the computations 4.1 The zeta function We recall the denition of the Euler Gamma function:

∞

Z

dt ts−1 e−t .

Γ(s) = 0

An innite product formula shows that this is never zero. It has a meromorphic continuation to

C

and has simple poles at non-positive integers. The residue at

s=0

is equal

1.

to

We would like now to consider the heat kernel expansion for the Laplace operator

4.

We renormalize it to take into account the zero eigenvalue:

Tr+ (e−t4 ) = Tr(e−t4 ) − dim ker 4. Incidentally, we know from (18) that

dim ker 4 = 1.

The

zeta function

of the Laplace

operator is dened as

1 ζ4 (s) = Γ(s) We are interested in evaluating

ζ(0)

Z

∞

dt ts−1 Tr+ (e−t4 ).

0

in two dierent ways, and compare the two results.

Notice that

X

Tr+ (e−t4 ) =

e−tλ

λ∈Spec(4) λ>0

Tr+ (e−t4 ) decays exponentially when t → ∞ (it is crucial to have subtracted the λ = 0 eigenvalue). Instead, there is an asymptotic series as t & 0 for Tr+ (e−t4 ) given so that by

+

−t4

Tr (e where

) ∼

t&0

∞ X

(n−2)/2 a+ n (4)t

n=0

( an (4) a+ n (4) = a2 (4) − dim ker 4

if if

n 6= 2, n=2

k ∈ N there exists an N = N (k) N X + −t4 + (n−2)/2 an (4)t Tr (e ) − ≤ Ck tk n=0

(compare (6)). By denition, this means that for every such that

for some constant

Ck .

Theorem 4.1. The function Γ(s)ζ4 (s) has a meromorphic extension to C with isolated

simple poles at s = (2 − n)/2 for n ∈ N, and Res s=(2−n)/2

Γ(s)ζ4 (s) = a+ n (4).

16

In particular ζ4 (0) + 1 = a2 (4).

Proof.

(23)

Write

1

Z Γ(s)ζ4 (s) =

dt t |0

∞

Z

−t4

+

s−1

dt ts−1 Tr+ (e−t4 ) . {z }

)+ } |1

Tr (e {z

=:I0 (s)

=:I1 (s)

The integral I1 (s) is exponentially convergent due to the decay property at innity of Tr+ (e−t4 ), so it denes an holomorphic function of s. Thus the only singularities can come from

I0 (s):

to evaluate it, we use the asymptotic expansion of the heat kernel.

Write

−t4

+

Tr (e

)=

N X

(n−2)/2 a+ + N (t) n (4)t

n=0 where the remainder

N (t)

is smooth in

t

and

|N (t)| ≤ Ck tk

if

N ≥ N (k).

Hence

I0 (s) = =

N X n=0 N X

a+ n (4)

Z

1 s+(n−2)/2−1

dt t

Z

0

a+ n (4)

n=0

1

+

dt ts−1 N (t) =

0

1 + EN (s) s − (2 − n)/2

and the modulus of the last summand can be estimated by

1

Z

s−1

dt t

Z

0 which is convergent if half-plane.

dt ts−k−1

0

<(s) > −k .

EN (s)

The function

is thus holomorphic in this

We use the uniqueness of the analytic extension to see that the latter is

independent of

k.

The last statement now follows from the fact that of

1

|N (t)| ≤ Ck

a+ 2 (4).

Res Γ(s) = 1 and from the denition s=0

4.2 The Connes-Tretko way to compute ζ(0) We now come to the second way of evaluating the value of + −t4 by denition of Tr (e ) we have

Z

∞ s−1

Γ(s)ζ4 (s) =

dt t 0

Z = 0

+

−t4

Tr (e

Z )=

∞

dt ts−1 Tr(e−t4 ) −

1 s

17

∞ s−1

dt t 0

ζ(0) [1].

Tr(e

−t4

First, we notice that

Z )− 0

∞

dt ts−1 =

and hence



 ˆ Res Γ(s)ζ(s) = Res Γ(s)ζ4 (s) − 1 s=0

(24)

s=0

where we have set

1 Γ(s)

ζˆ4 (s) =

∞

Z

dt ts−1 Tr(e−t4 ).

(25)

0

We can rewrite the trace of the heat kernel as an integration in symbol space. Indeed, by denition of heat kernel one should have

e

−t4

Z

dx k(t, x)αx (a). (2π)2

a= R2

Comparing this with (20), we are lead to dene

Z

dξ e−ix·ξ σ(e−t4 )(ξ).

k(t, x) = ˆ2 R By denition, the trace of an operator

A

Tr(A) =

acting on the Hilbert space

X

H0

is given by

hAeν , eν i

ν∈N where

Aθ

{eν }ν∈N

H0 . As this space {eν = U n V m }(n,m)∈Z2 .

is a complete orthonormal system in

via the GNS construction, we may choose

H0 ,

the denition (17) of the scalar product in

X

Tr(A) =

is obtained from Moreover, using

we obtain

τ ((U n V m )∗ A(U n V m )) .

(n,m)∈Z2

A = e−t4 we  Z X n m ∗ τ (U V )

Applying these considerations to

Tr(e

−t4

)=

R2

(n,m)∈Z2

=



X

get

τ

n

m ∗

Z

(U V )

R2

(n,m)∈Z2

 dx n m k(t, x)αx (U V ) = (2π)2  dx ix·(n,m) n m k(t, x)e U V (2π)2

n m where we have used the denition (16) of αx (U V ). Using the cyclicity of the trace, n m ∗ n m we may move (U V ) at the end of the argument of τ , and this cancels with U V because these are unitaries. Moreover, we can bring the series inside the argument of by linearity of the trace, which yields to



 Z

Tr(e−t4 ) = τ 

R2

dx k(t, x) 

 1 (2π)2

X (n,m)∈Z2

Recall now the Dirichlet kernel formula

1 (2π)2

X

eix·(n,m) = δ(x)

(n,m)∈Z2

18

eix·(n,m)  .

τ

from which we deduce that

Tr(e

−t4

Z

 Z dx k(t, x)δ(x) = τ (k(t, 0)) =

)=τ

dξ τ (σ(e−t4 )(ξ))

(26)

ˆ2 R

R2

τ

where we have used again linearity of the trace to commute

with integration in

dξ .

Next we use Cauchy formula

e where

γ

−t4

1 = 2πi

Z

dλ e−tλ (4 − λ)−1

γ

is a curve encircling the non-negative real axis in the complex plane, oriented e−t4 is thus

in the counter-clockwise fashion. The symbol of

−t4

σ(e

1 )= 2πi

Z

dλ e−tλ σ((4 − λ)−1 ).

γ

Plugging this and (26) into the denition (25) of

1 ζˆ4 (s) = Γ(s)

Z

∞ s−1

Z

Z

dt t

dξ ˆ2 R

0

γ

ζˆ4 (s),

we obtain


dλ −tλ e τ σ((4 − λ)−1 )(ξ) . 2πi

We exchange the order of the integrals and observe that

Z

∞

Z

s−1 −tλ

dt t

e

t

0

∞

λ−1 dt λ−(s−1) ts−1 e−t = λ−s Γ(s)

=

tλ

0

to conclude that

ζˆ4 (s) =

Z γ

dλ −s λ 2πi

Z


dξ τ σ((4 − λ)−1 )(ξ) .

ˆ2 R

In particular we obtain

Res Γ(s)ζˆ4 (s) = s=0

Z γ

dλ 2πi

Z

−1


dξ τ σ((4 − λ) )(ξ) = Res λ=0

ˆ2 R

Z


dξ τ σ((4 − λ)−1 )(ξ)

ˆ2 R

by the residue theorem. In order to evaluate this residue, we have to nd the coecient −1 of λ in the Laurent expansion of the function on the left-hand side. We use Theorem 3.1 to nd a parametrix

R(λ)

to approximate the resolvent of

4 − λ:

it turns out that

we will only need the rst three terms in the homogeneous expansion of its symbol, so we write

σ(R(λ))(ξ) ∼2 b0 (ξ, λ) + b1 (ξ, λ) + b2 (ξ, λ). Recall that each

bj

is homogeneous in

(ξ, λ)

of order

−2 − j ,

that

bj (νξ, ν 2 λ) = ν −2−j bj (ξ, λ)

19

which means by denition

(λ is considered a symbol of order

2,

the order of

4).

From this we deduce that for

j=2 Z Z 1/2 −4 dξ τ (b2 (ξ, λ)) = ((−λ) ) ˆ2 R

=

ξ (we set

λ = −1

(−λ)−1/2 ξ


dξ τ b2 ((−λ)−1/2 ξ, −1) = ˆ2 R Z Z −1 −2 (−λ)dξ τ (b2 (ξ, −1)) = −λ (−λ)

ˆ2 R

ˆ2 R

because it is in the resolvent of

4!).

dξ τ (b2 (ξ, −1))

Putting this result together with

Equations (23) and (24), we conclude that

a2 (4) = ζ4 (0) + 1 = Res Γ(s)ζˆ4 (s) = −

Z

s=0

ˆ2 R

dξ τ (b2 (ξ, −1))

which is [1, Equation (25)].

References [1] Alain Connes, Paula Tretko,

The Gauss-Bonnet theorem for the noncommutative

two torus. arXiv:0910.0188v1 [math.QA] (2009).

Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. CRC Press, 2nd edition (1995).

[2] Peter B. Gilkey,

[3] José M. Gracia-Bondía, Joseph C. Várilly, Héctor Figueroa,

mutative Geometry. Birkhäuser (2001).

[4] Henry P. McKean, Isadore M. Singer,

Elements of Noncom-

Curvature and the eigenvalues of the Lapla-

cian. J. Di. Geometry 1 (1967), 4369.

[5] Steven Rosenberg,

The Laplacian on a Riemannian Manifold. Cambridge University

Press (1997).

20