Noncommutative Geometry

Nigel Higson Penn State University Noncommutative Geometry

Alain Connes

1 What is Noncommutative Geometry?

Geometric spaces approached through their algebras of functions.

The spaces are often very singular (defined by equivalence relations, or even groupoids).

The function algebras are typically noncommutative.

The algebras/spaces are analyzed using Hilbert space tools.

In particular, spectral properties of algebras, viewed as algebras of operators on Hilbert space, are crucial. One might call the subject spectral geometry.

2 What are its Origins?

Werner Heisenberg

What Heisenberg understood . . . is that [the] Ritz-Rydberg combination principle actually dictates an algebraic formula for the product of any two observable physical quantities . . .

3 Heisenberg wrote down the formula for the

product of two observables;

¡£¢ ¤¦¥¨§ ©   ¢ § © ¤§   and he noticed of course that this algebra is

no longer commutative,

¢ ¤   ¤¢ 

. . . The right way to think about this new phenomenon is to think in terms of a new kind of space in which the coordinates do not commute. The starting point of noncommutative geometry is to take this new notion of space seriously.

Alain Connes Noncommutative geometry, Year 2000

4 Commentary of Riemann

. . . it seems that the empirical notions on which the metric determinations of space are based . . . lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry; and in fact one ought to assume this as soon as it permits a simpler way of explaining phenomena.

Bernhard Riemann On the Hypotheses which lie at the Foundations of Geometry

5 What Has Noncommutative Geometry Accomplished?

Manifold topology (progress on the Novikov conjecture, Gromov-Lawson conjecture, etc).

Harmonic analysis, especially of discrete groups.

Models in physics (notably of the quantum Hall effect).

Foliation theory and Atiyah-Singer index theory, on singular spaces, or parametrized by singular spaces.

In addition, NCG may offer the prospect for progress in fundamental physics, arithmetic, . . .

6 Spectral Theory and Hilbert Space

David Hilbert in 1900

In the winter of 1900-1901 the Swedish mathematician Holmgren reported in Hilbert’s seminar on Fredholm’s first publications on integral equations, and it seems that Hilbert caught fire at once . . .

Hermann Weyl David Hilbert and his mathematical work

7 Helmholtz Equation

Hilbert saw two things: (1) after having

constructed Green’s function for a given

 $ !#" region and for the potential equation

. . . , the equation

 $

! % & '(% for the oscillating membrane changes into a

homogeneous integral equation

¡*)+¥ ¡*)-,/.0¥ ¡.0¥213.  $

% & ' %

¡£.-,4)+¥  ¡*)-,/.+¥

with the symmetric , . . . ;

(2) the problem of ascertaining the “eigen

¡*)+¥ % values” ' and “eigen functions” of this integral equation is the analogue for integrals

of the transformation of a quadratic form of 5 variables onto principal axes.

Hermann Weyl David Hilbert and his mathematical work

8 Problem of H.A. Lorentz

. . . there is a mathematical problem which will perhaps arouse the interest of mathematic- ians . . . In an enclosure with a perfectly reflecting surface there can form standing electromagnetic waves analogous to tones of an organ pipe . . . there arises the mathematical

problem to prove that the number of sufficiently

1

6 7 6 high overtones which lie between 6 and is independent of the shape of the enclosure and is simply proportional to its area.

H.A. Lorentz Wolfskehl Lecture, 1910

9

Reformulation



!#"98 ':8+";8

 $

8=A@

"

¡ ¥C

B

! '

' # eigenvalues of less than or equal to .

¡ ¥ ¡ ¥

B

 

' Area

DFEHGlim

' constant

This is equivalent to the asymptotic relation

  8

' constant

¡ ¥

limEHG

8 5 Area

10 The idea was one of many, as they probably come to every young person preoccupied with science but while others soon burst like soap bubbles, this one soon led, as a short inspection showed, to the goal. I was myself rather taken aback by it as I had not believed myself capable of anything like it. Added to that was the fact that the result, although conjectured by physicists some time ago, appeared to most mathematicians as something whose proof was still far in the future.

Hermann Weyl Gibbs Lecture, 1948

11

Compact Operators

JK L K Definition. A bounded linear operator I on a Hilbert space is compact if it maps the closed unit ball of Hilbert space to a (pre)compact set.

Example. If I is a norm-limit of finite-rank operators

then I is compact.

Elementary calculus M the maximum value of the

¡POQ¥R S OTSVU

I K function N on the closed unit ball of is

an eigenvalue for IXWYI .

JZK L K Theorem (Hilbert et al). If I is a compact and selfadjoint operator then there is an

orthonormal basis for K comprised of eigenvectors

for I . Thus

'^]

`

U



'

`

I [ \

':_ \

... ]

Theorem (Rellich Lemma). !ba is a compact operator.

12

Spectral Theory for the Laplacian

 e e

c d

W !

The Laplace operator.

U

¡ ¥ c

Theorem. There is an orthonormal basis for f

consisting of functions gF8 for which



!hgV8 ':8igV8

in the distributional sense. The eigenvalues 'j8 are

positive and converge to infinity.

¡ ¥

Spec !

G

¡ ¥

c

k l Remark. In fact one can show that gF8 . This follows from elliptic regularity.

13

Singular Values

¡ ¥n, ¡ ¥n,opq

U

]

I m I Definition. The singular values m of

a bounded operator I are the scalars

S OTS

¡ ¥r 

I

§srut

SYOTS s I

mi8 inf sup

] 8

dim vnw

a

¡ ¥x ¡ ¥yx qz

U

I m I

Observe that m^] and that

¡ ¥r $2

m I { I 8

is compact limEHG 8

Now let I be compact, self-adjoint, and positive

O},£O=~ x $ ¡ ¥

| I ' I (meaning ). List the eigenvalues 8 in decreasing order, and with multiplicity.

Theorem. If I is compact, self-adjoint, and positive

¡ ¥r ¡ ¥

I ':8 I

then m8 .

]

'

`

U



'

` I

Proof. \

'

_ \ ...

14 Trace Class Operators

Lemma.

¥ ¡ ¥€ ¡ ¥V ¡ ¥€ ¡

U U U U

] ] ]

7 mi8 I m 8 I 7 I mi8 I 7 I mi8 I

¡ƒ‚ ¥n, ¡ ‚Q¥y€ S„‚TS ¡ ¥o

mi8 I mi8 I mi8 I

¡ ¥ K

Definition. The trace ideal in is

¡ ¥C ‡ ¡ ¥Qˆ ‰ Š=

†

]

<

K I mi8 I

¡ ¥

†

]

I k

Definition. If K then

G

 

¡ ¥r O , O ~„

I | I

Tr Pt ]

The sum is over an orthonormal basis. Note: if

O ,qzŒ,*OjŽ ‹

] is an orthonormal set then

 

O , O ~ € ¡ ¥V

8 8 < < 8

| I m I

t t

] ]

8 8

15

As with the usual trace,

‚ ¡ ¥F, ¡ ¥ ¡ƒ‚ ¥ ¡ ‚Q¥V

†

]

K I k K M I I

k Tr Tr

c ‘ c

Example. If  is smooth on and if

¡’}¥C “ ¡’,•”H¥ ¡P”¦¥1–”—,

9g  g

then is a trace-class operator, and

“

¡˜’y,™’}¥1’} ¡ ¥y  Tr

16

Dixmier Trace

G

¡ ¥rˆ ‰  ¡ ¥C

†

]

8

I š I K 5 ›im

Definition. sup 8

š

š

š

G

¡ ¥œ ¡ ¥œ ¡ ¥

† †

] ]

K K

Observe that K .

G

¡ ¥

†

]

I k ž Definition. If K is positive and LIM is a

Banach limit then define

¡ ¥ ¡ ¥ƒ



¡ ¥„

ž ž Ÿ

B mi8 I I

Tr LIM 8¡

log

 ¡ ¥V



¡ ¥

Ÿ

ž B ' I 8

LIM 8:¡ log

Theorem (Dixmier). If LIM ž has the property

¡Y¢ ,£¢ ,¢ ,pqz¥C ¡/¢ ,¢ ,¢ ,¢ ,£zqp¥

U U U

ž ] ž ] ]

LIM _ LIM

¡ ¥y ¡ ¡ ¥ ¥

U U

ž ž ž ] ]

I 7 I I I then Tr Tr 7 Tr .

17 Integration

Now back to Weyl’s

c d

Theorem . . .

¡ ¥ ¡ ¥

c

! [ ¤ › 5 ¥ ¦ ':8 vol

Weyl’s Theorem shows that

¡ ¥C ¡ ¥o

c

¦

a

ž

¤ ›

Tr ! vol

¥

c

L §

More generally, given g3J we get

¡ ¥C “ 1 

¦

a

ž

¤ › g

Tr g¦›Œ! vol

¥

¦ a

This suggests that ! is some sort of ‘volume c element’ for the manifold¥ . . .

18

Spectral Triples

¡£¢ , ,V¨ ¥

Definition. A spectral triple is a triple K

consisting of a separable Hilbert space K , an ¢

algebra of bounded operators on K , and a ¨

(typically unbounded) selfadjoint operator on K ,

for which:

U

¡Y© ¨ ¥

]

a

the operator 7 is compact, and

¢ ¨ ,  ¨ ¨

k « ªi¬ ª & ª if ª then the commutator

extends to a bounded operator on K .

According to Connes, spectral triples constitute an extension of the notion of Riemannian geometric space which is broadly applicable to problems in

fundamental physics, number theory, . . . .

U

¢ ¡A© ¨ ¥

] a

Remark. If has no unit, replace 7 with

U

¡A© ¨ ¥

]

a 7 ª in the above.

19 The Standard Example

Basic ideas:

¨

Regard as a ‘square root’ of ! .

¨ , ¢

« ª k Think of ªi¬ as a gradient of .

The simplest case is

1

G­

U

¢   ¡® ¥F,  ¡® ¥F, ¨ 

¯

1i’

Ÿ

l K f

& Ÿ

The theory of Dirac-type operators in geometry provides further ‘commutative’ examples in the

context of Riemannian manifolds.

U

¨  1 1 ¨  e e

W W

7 on forms

U

¨  ¨  e e

Dirac Operator W on spinors

20 The Operator F

Write

U

 ¨ ¨  ° ,

! ±

! and ¥

so that

U

©4 °  ° ° W and [

In the simplest case this is the Hilbert transform:

¡P”¦¥

° ¡’}¥C 1–”H

g

’ ”

Ÿ

g

²r³ ´

& °

The operator is important! ¨

Roughly speaking the distinction between and ! ± corresponds to the distinction between densities and¥ differential forms (on manifolds).

21 Groupoids and Quotients

Let µ be a smooth etale´ groupoid: µ

range source

¡ ¥

Obj µ

(source and range are local diffeomorphisms).

­

G

¢  ¡ ¥ l

Let µ and define

¡¸·h¥C ¡¸· ¥ ¡¸· ¥V

U U U

]i¶ ] ]

g g g g

t

¹ ¹ ¹

¥ ±

We obtain the convolution algebra of µ .

22

Examples

» ¼ ½¿¾ÁÀÃÂ

Heisenberg example, º

» ÄoÅHÆÈǦÉÅCÇ » Ê Ç^ÅZÑ

Kronecker foliation, º ¥ÌËÎÍÐÏ

23

Infinitesimals and Differentials

Definition. An infinitesimal of order  is a compact

¥ ¡ ¥ Ò ¡

a

m8 I 5

operator I for which .

¢ 1  °Ó,

k ª « ªi¬ Definition. For ª define .

Example. In the basic case,

¡’}¥ ¡P”¦¥

1 

g & g

’ ”

g operator with integral kernel , &

give or take a factor of ²r³ .

°Ó, qp °Ó,

]

ª ¬ « ªiÖ¬ If we grade differentials ª=ÔÕ« according

to degree and use the graded commutator then

U× Ø

1 $=, as in de Rham theory.

24

Connes’ dictionary

¡Ì¢ , ,V¨ ¥

Geometric space Spectral triple K ¢

Complex variable Operator in ¢ Real variable Selfadjoint operator in

Infinitesimal Compact operator

1 °Ó,

« ªi¬

Differential ª Commutator

¡ ¥

ž

Ù ª Integral Dixmier trace Tr ª

. .

. .

25

Zeta Functions

) Ú

d

U a:Û Theorem. If then ! is

a trace-class operator. ÜÞÝ

Proof. Follows from Rellich Lemma.

Theorem (Minakshisundaram and Pleijel). The

zeta function ß

¡ ¥ ¡È)à¥C a(á

Tr ! ¥

is meromorphic on § with only simple poles. §

Residues Vanish Actual Poles

Singularities of âz¾äãå for a closed surface.

26

Weyl’s Theorem §

Residues Vanish Actual Poles

Abelian-Tauberian Theorem. Let I be a positive, a:Û

invertible operator and assume that I is trace-

) Ú

class for all . Then

Ÿ

¡ ¥

B æ

 ¡*) ¥ ¡ ¥  

'

a:Û

l { & I l

DFEHGlim lim Tr

]

Ÿ

' ۃç

See Hardy, Divergent Series. The theorem says

 ¡È)   ¥

a:Û

Ÿ

' l { & l 8

DFEHGlim lim

]

Ÿ Ÿ

'

D/è D

8

Ûƒç ¡

27 The proof of meromorphicity uses pseudodifferential

operators

è

‚ , 

áPê

a

Ié! I 5

[ differential of order ¥

Lemma (Guillemin). Suppose that for every holo-§©ì

‚

§ © ë

morphic family there are pseudodifferential ] ,

Û

í ]

and î such that

Û

Û a

§©ì

§ ©

¡Y1 )+¥ï‚  , 

í

7 «ðë ¬27 î ]

]

©

Û Û a

Û

¡ƒ‚ ¥

Then Tr is meromorphic, with simple poles.

Û

¡È)à¥Tñ $

Proof. If Re then

§©ì

§ ©

¡•‚ ¥C ¡ , ¥ ¡ ¥™ó

í

©

1 ) ò

Ÿ ]

«ðë ¬ 7 î

Tr Tr ] Tr

Û Û a

Û

7

 ¡ ¥

1 )

Ÿ ]

Tr î

Û a

7

¡ƒ‚ ¥y ¡/1 )ॠ¡ ¥

]

]

a î

Hence Tr 7 Tr .

Û Û a

28

¡ƒ‚ ¥ 1ô, 1ô,/õ 1ô,qz

& &

The poles of Tr are located at & .

Û

Ÿ

§

1

1

&

&

Ÿ

¡•‚ ¥

Domain of Tr

Û

¡ ¥ ]

Domain of Tr î Û a

29 ¨

Lemma. If is (pseudo)differential of order 5 then

d

©

¨ ,Β  ¨ ,

©

’

« ¬ ö 5 & î

©t

]

ö

5 &

where î has order , and hence

Ÿ

d d

÷ ÷

© ©

¡/1 ¥ï¨  ¨ ,ï’ ’ ¨ , 

©Èø ©™ø

’ ’

7 5 & 7 î

ö ö

©t ©t

] ]

ö ö

Proof. This follows from the Heisenberg relations

÷

 ©¸

,ï’  ù ©F

ø

©

’

ö ö

As a result, Weyl’s Theorem follows from Guillemin’s Lemma.

30

Cyclic Cocycles

¡£¢ , ,4¨ ¥

Let K be a spectral triple.

Proposition. The formula

¡/ú °Ó, °Ó, ¥ ¡ , ,qpŒ, ¥C

] ]

8 8

Ô Ô

ª « ª ¬û›z›q›Ó« ª ¬ N ª ª

ª Tr ¢ defines a multlinear functional on with the

following properties:

¡ , ,£zqÕ, ¥C ¡ ¥ ¡ ,qzŒ, , ¥

] ]

8 8 8

N ª ª ª & N ª ª ª

Ô Ô

Ÿ

¡ ,£zqÕ, ¥C $

]

80ý

ü

ªjÔ ª

N , where

¥ ¥C ¡ ,qpŒ, ¡ ,qpŒ,

] ] ]

80ý 80ý

ü

Ô Ô

N ª ª ª N ª ª

U

¡ , ,qzŒ, ¥

] ]

80ý

Ô

& N ª ª ª ª

pq

7

¡ ¥ ¡ ,£pqþ, ¥V

] ]

8+ý 80ý 8

Ô

7 & N ª ª ª Ÿ

31

Cyclic Cohomology 5

Lemma. Let N be a cyclic -linear functional. Then

¥ ¡

ü

5 7

N is a cyclic -linear functional, and

Ÿ

U

 $

ü

N . ¢

Definition. Let be an algebra. The 5 th cyclic ¢

cohomology group of is

5 ¥y

¡Ì¢ cyclic -cocycles 8

K l modulo cyclic coboundaries.

The cocycle N is a sort of ‘fundamental class’ for the

¡£¢ , ,V¨ ¥ spectral triple K . It reflects information from

index theory:

è

¡äÿ¦,™ÿH,ppÕ,Èÿ#¥o ¡ ÿT°zÿ#¥T ¡ ¥ N

Index &

¥ Ÿ

32 Continuation of the Dictionary

. .

1 °Ó,

« ªi¬ Differential ª Commutator de Rham theory Cyclic cohomology

. .

33

Local Index Formula

¡£¢ , ,V¨ ¥

Theorem (Connes and Moscovici). Let K be a spectral triple with simple dimension spectrum.

The local formula

¡ ,qzŒ, ¥C

¡

Ô Ö

ª ª

¤

Ö

§ð  §ð ¥¤

 t

ó

ú ¨ , ¨ , ,

]

ò a a§¦ ¦ a:Û

Ô Ö

ª « ª ¬ ›q›p›à« ª ¬ ! ¤

£¢

Res Tr

±

Û

Ö Ô

¥ Ô

where

¡ ¥

¡ ¥

Ö

U



 < <

 7

¦ ¦

&

qp ¡ ¥„¡ õ ¥ ¡ ÿX¥

Ÿ

¤ ›

U

Ö

]©¨ ¨ ]

   7  7 ›q›p›  7

Ö Ö Ÿ defines a cocycle which is cohomologous to the

fundamental cocycle N .

§ð 

U U U

 £¨ ,§ ¨ ,qp ¨ , qp 

« I ¬ Notation. I .

34 Comments

The formula is a starting point for Atiyah-Singer index theory in noncommutative geometry.

In the classical case (Riemmanian manifolds) the

residues are computable from the coefficients of ¨

(Seeley, Wodzicki, et al). ¨ Small (smoothing operator) changes in leave the index formula invariant.

In the noncommutative world, local means ‰ concentrated at in momentum space (c.f. Fourier theory).

35