WHAT IS NONCOMMUTATIVE GEOMETRY Dr. FATIMA M. AZMI

WHAT IS NONCOMMUTATIVE GEOMETRY
BY
Dr. FATIMA M. AZMI
In the early 80's, Alain Connes came up with the idea of generalizing the notion of space and this led to the birth of noncommutative geometry which came as set of tools and methods
to deal with problems that was beyond the reach of classical
methods.
Example
- M a compact Hausdor® space.
- ¡ a discrete group acting on M .
The quotient topology on the orbit space M=¡ may not seperate
orbits in M=¡.
To over come this problem, Alain Connes' key observation is
that one can attach a non commutative algebra through non
commutative quotient space that captures most of the information.
His motivation came from one of the most important theory in
functional analysis, which shows the duality between commutative algebra and geometry.
In the late 40's two Russian mathematician Gelfand and Naimark
proved this important theorem in functional analysis.
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Gelfand - Naimark Theorem :
1- Let X be a compact Haudor® space, then C(X) is a commutative C ¤ algebra with norm jjf jj = supx2X jf (x)j < f 2
C(X) and f ¤(x) = f¹(x).
2- Given any commutative C ¤ algebra A one can construct a
unique compact Hausdor® space X such that A can be identi¯ed
with the algebra C(X).
f compact Hausdor® space Xg () f comm. C ¤ algebra C(X)g
Conclusion: Studying commuative C ¤ algebra amounts to
studying compact Hausdor® spaces and vice vers.
Connes' Proposal: Studying non commutative C* algebra amounts to studying non commutative spaces.
The noncommutative approach for studying the orbit space
M=¡ is as follows:
1. replace M by the C ¤-algebra C(M )
2. replace the orbit space M=¡, not by a smaller algebra
C(M=¡) but by a larger algebra, the crossed product algebra
C(M ) £ ¡ = ff =
X
g2¡
fg g; wherefg 2 C(M )andg 2 ¡g
It is a noncommutative algebra, and after completion in
some fashion it provides a powerfull tool for the study of
M=¡.
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2-NONCOMMUTATIVE MANIFOLDS:
- M a smooth Riemannian manifold
- C 1(M ) a commutative * algebra which is not a C ¤ algebra
and it lives on a manifold
Other useful geometrical objects on M de¯ned in terms of the
algebra C 1(M ) are for example:
- Â(M ) = the collection of all smooth vector ¯elds on M .
Note that Â(M ) is a C 1(M ) module. Moreover, any vector
¯eld X 2 Â(M ) gives rise to a di®erential operator acting on
smooth functions
X : C 1(M ) ! C 1(M ); f ! Xf
and it satis¯es the Leibniz rule;
X(f g) = X(f )g + f X(g); 8f; g 2 C 1(M )
Observation:
f vector ¯elds over M g () fderivations of the algebra C 1(M )
- di®erential forms, general tensor ¯elds, curvature, connection
and covariant derivative all these can be de¯ned purely algebraically from C 1(M). Thus they all can be studied in this
algebraic context.
The algebra C 1(M ) indeed characterizes the manifold M completely as shown by the following theorem.
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Theorem :
Two smooth manifolds M and N are di®eomorphic () the
algebra of functions C 1(M ) and C 1(N) are isomorphic.
Conclusion: All the di®erential geometric properties of the
manifold are encoded in the algebra C 1(M ).
Let A be a noncommutative algebra which is closely related to A == C 1(M ). Then one tries to mimick the various
de¯nition, based on A. This is a fruitful idea, as many of the
ordinary de¯nition still make sense in ths noncommutative algebraic context. The de¯nition of vector ¯elds as derivations
of A works quite well, as do the de¯nition of di®erentail forms
with exterior derivative, also tensor ¯elds, and so on.
One gets this way noncommutative vector ¯elds, di®erential
forms, tensor ¯elds, etc.
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3- NONCOMMUATIVE VECTOR BUNDLES AND
K THEORY
- X is a compact Hausdor® space.
- E is a complex vector bundle over X.
- ¼ : E ! X is a bundle map
- P = ¡(E) = fs : X ! E; ¼ s = idX g
Then P is a ¯nitely generated projective C(X) module. Also
conversly one can do the reverse.
The Serret- Swan Theorem:
fvector bundles on Xg Ã! f ¯nitely generated proj. C(X) moduleg
Conclusion: Instead of studying vector bundles over compact Hausdor® space, one can study ¯nitely generated projective modules over commutative C ¤ algebra.
Moreover one usally thinks of ¯nite projective modules over
noncommutative algebra as noncommutative vector bundles.
Thus, the idea in noncommutative geometry is to treat certain
classes of noncommutative algebra as noncommutative spaces
and try to extend tools of geometry, topology and analysis to
this setting. It should be emphasized, however that as a rule
this extension is never straight forward and always involve new
phenomena.
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In the commutative world the power tools are homology and the
fundamental group. They don't have a straightforward generalization to the noncommutative world.
On the otherhand topological K-theory (which classi¯es vector
bundles ) is the most powerfull tool that passes immediatly to
the noncommutative world.
Swan's theorem states that
K0(C(M )) = K 0(M ); for M a compact Hausdor® space
In the frame work of noncommutative geometry, one can regard
the elements of the group K0(A), where A is a noncommutative
algebra of functions on noncommutative space as vector bundles
over the noncommutative space.
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Chern -Connes character:
The classical Chern character is a natural transformation from
K- Theory to ordinary cohomology theory with rational coe±cients.
Ch : K 0(X) ! ©i¸0H 2i(X; Q); X is compact Hausdor® space
When X is smooth manifold there is an alternative construction
of the map Ch, called the Chern - Weil construction, that uses
the di®erential geometric notions of connection and curvature
on vector bundles.
- E is a complex vector bundle on X
- r is a connection on E
- R the curvature form.
The Chern character of E is then de¯ned to be the class of non
homogeneous even form
Ch(E) = T r(eR )
With all the advances made on the K-theory, the lack of cohomological companion to K-theory and an e®ective computational devise remained a serious obstruction, and noncommutative topology was now in need of its own homology theory
which would allow making concrete calculations.
The breakthrough came in 1981 with the discovery by Alain
Connes of cyclic cohomology and of spectral sequence relating
it to Hochschild cohomology .
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4- QUANTIZED CALCULUS:
With the translation of the problem from geometry to algebra
and analysis. The need of new calculus appeared. Connes
developed a calculus which would replace the usual di®erential
and integral calculus.
The following dictionary shows the noncommutative analogues
of some of the classical theories and concepts originally conceived for spaces:
CLASSICAL NOTION OF CALCULUS
Complex variable
Real variable
In¯nitesimal
P @f
Di®erential of f is df = @x
dxi
i
Integral of in¯nitesimal of order 1
QUANTIZED CALCULUS
Operator in H
Self adjoint operator in H
Compact operator in H
df = [f; F ]
Dixmier trace, T r(T )
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Fredholm Module
Let A be a unital * algebra, an odd Fredholm module is a triple
(H; ¼; F ) where H is a seperable Hilbert space on which A acts
by a ¤ representation ¼ and F = F ¤ is a bounded operator on
H such that F 2 ¡ I and [F; ¼(a)] are compact operators for all
a 2 A.
A Ferdholm module is called even if in addition the Hilbert
space is Z2 graded.
Key Operator : Dirac Operator
- M a smooth compact oriented Riemannian manifold of dimension 2n
- E a spinor bundle over M .
The Dirac operator D is a ¯rst order di®erential operator
D : L2(E) ¡! L2(E)
Locally in terms of a local orthonomal frame ei of T M ,
D=
2n
X
i=1
ei ² rei ; where
- r is the covariant deriv. on E determined by the connection.
- \:" denotes Cli®ord module multiplication, where the ei 's satisfy the Cli®ord multiplication relation
ei:ej = ¡ej :ei ; and e2i = ¡1
There is a natural inner product for the spinor bundle, which
turns the L2 sections of E on M into a Hilbert space and the
algebra C 1(M ) acts on the Hilbert space. Connes calls the
system (H; C 1(M ); D) a spectral triple.
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SOME APPLICATIONS
Applications Include the Following Areas:
1. Index theory of elliptic operators.
2. Algebraic and di®erential topology.
3. Generalization of Riemannian manifold
4. Deformation quantization and quantum geometry
5. Number theory
6. Theory of Foliation
7. Quantum ¯eld theory, and Theoretical physics.
1 - INDEX THEORY
Classi¯cation of manifolds is an important thing. Theory of homology and cohomology was developed to help classify spaces.
For example, the torus T 1 has genus one (one hole) where as
the sphere S 2 has genus zero (no hole). Thus T 1 is not di®eomorphic to S 2.
Let M be a manifold with genus k. The Euler charactristic
Â(M ) is given by
Â(M ) =
X
(¡)pdimH p(M; C) = 2 ¡ 2k:
Thus , Â(S 2) = 2 and Â(T 1) = 0.
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As we go higher in dimension the spaces gets more complicated,
and we need new tools to compute invariants. Invariants of
manifolds can be studied through invariants of certain type of
di®erential operators, if we do so, then we have moved the
problem from topology to analysis and geometry.
The concept of index associated to certain type of operators,
arose in functional analysis. It came as a study of the relation
between analytic and topological invariants of a certain class of
linear maps between certain spaces.
Index of operator:
- M a smooth compact oriented manifold
- E, F complex vector bundles over M
- d an elliptic linear di®erential operator
d : ¡1(M; E) ¡! ¡1(M; F )
Then, ker d and coker d are ¯nite dimensional. The Fredholm
index of d is
ind(d) = dim(kerd) ¡ dim(cokerd) 2 Z
The index of an elliptic operator is a stable object , it
remaines constant under continuous perturbation, i.e it depends
only on the homotopy class of the operator.
As Dirac operator is an elliptic operator, thus it has an index,
which is homotopy invariant.
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Example:
@
is the Dirac operator on L2(S 1),
Let M = S 1 and D = i @µ
and ind(D) = 0.
In 1963, Atiyah and Singer in their famous theorem expressed
the index formula for elliptic di®erential operator over smooth
compact manifold M in terms of topological data related to M
and its curvature.
Theorem
- M be a smooth compact Riemannian spin manifold of even
dimension
- E a spinor bundle over M .
- D is the Dirac operator
D : ¡1(M; E) ¡! ¡1(M; E)
Then the Atiyah-Singer Index theorem takes the form
ind(D) =
Z
M
^ ):
A(M
A^ genus is a charactristic class which involves the curvature,
it is a di®erential invariant
Theorem:
Let M be a compact spin manifold of dimension 4n. If M
^ ) = 0.
admits a metric of positive scalar curvature, then A(M
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Chern - Connes character
Let (H; ¼; F ) be an even Fredholm module over an algebra A.
Associated with this Fredholm module is a fundamental cyclic
cocycle class Ch(F ), called the Chern character. It is given by
Ch(F; H)(a0; a1; :::; an) = T r(°a0[F; a1] ¢ ¢ ¢ [F; an]):
This general concept of the Chern character leads to an index
theorem which is an extension of the classical Atiyah-Singer
index theorem.
Remark: Consider the following data:
- M is a compact spin manifold with spinor bundle E
- G is a discrete group which acts properly on the manifold , and
A = C ¤(G; M ) is a smooth crossed product algebra completed
in certain norm.
- H = L2(E)
- D is an odd Dirac operator on H
The equivariant Chern- Connes character of the module (D; H)
as an entire cyclic cocycle is given by;
X
((Ch(D; H); ( fg0 g0; : : :
=
=
X
Z
g0
g0 ;:::;g2k ¢2k
X
X
X
g2k
fg2k g2k ))
2
2
T rs (fg0 g0e¡t1 D ¢ ¢ ¢ [D; fg2k g2k ]e¡(1¡t2k )D )dt1 ¢ ¢ ¢ dt2k
g2G g=g0 ¢¢¢g2k
½X Z
j
(Fg )j
[L2k (g)]j
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¾
CONNES' VERSION OF NONCOMMUTATIVE
GEOMETRY :
1- Geometry for Connes is metric geometry, i.e the study of
manifolds with a Riemannian structure given by a metric tensor.
2- He considers compact manifolds of arbitrary dimension with
Riemannian structure, which gives rise to a ¯rst order di®erential operator, the Dirac operator. He shows that the manifold,
including the metric tensor can be reconstructed from the discrete eigenvalues of this operator. He encodes the properties of
the spectrum in a mathematical object called by him a spectral triple, which conatins a number of data and completely
describes the Riemannian manifold. Thus formulating the ordianry Riemannian geoemtry as a commutative Riemannian
geometry.
COMMUTATIVE SPECTRAL TRIPLE :
Consider the spectral triple (H; A; D) as de¯ned above, where
M is an n dimensional compact spin manifold, and the Hilbert
space H is formed from the spinor bundle E over M and A =
C 1(M ).
Connes exhibits a number of algebraic properties of this system,
characterizing the various characteristics of the manifold and
the metric and puts them in a list of seven properties. He
then proves a theorem which might be called the \ GelfandNaimark theorem for compact Riemannian manifolds and
spectral triple ".
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THEOREM :
1- For every compact spin manifold (M; g) there is an associated
spectral triple (H; A; D) as de¯ned above.
2- For every spectral triple (H; A; D) with H a Hilbert space, A
is a commutative algebra of operators in H, and D is a linear operator satisfying the seven properties, then there exist a unique
compact manifold (M; g) with metric g such that (H; A; D) is
the spectral triple associated with (M; g). Moreover the manifold M and the metric tensor can be constructed in an explicit
way from (H; A; D)
The notion of the spectral triple can be generalized to a version
in which one has a noncommutative algebra A .
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