Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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map 

: K



) ! K



(þ) ﬃ Z. Then K



(M) is defined

to be ker(

), also called the reduced K-theory. If X

is a closed subset of X, the K-theory of the pair

(X, X

) is defined as the reduced K-theory of the

quotient space X= X

. A fundamental computation

of Bott is the computation of the K -theory of

Euclidean space, K

) ﬃ Z with canonical gen-

erator called the Bott class b 2 K

), and

n1

) = {0}.

Some of the basic properties of K-theory are listed

as follows. Details can be found in Karoubi (1978).

1. Pullback If f : N ! M is a continuous map, then

given a vector bundle  : E ! M over M,the

pullback vector bundle is defined as f



(E) = {(x, v) 2

N  E : f (x) = (v)} over N. This induces a pullback

homomorphism, f

: K



(M) ! K



(N).

2. Push-forwa rd Let f : N !M be a smooth proper

map between compact manifolds which is

K-oriented, that is, TN  f



TM is a spin

vector

bundle over N. Then there is a pushforward

homomorphism, also called a Gysin map,

: K



(N) !K

þd

(M). where d = dim M  dim N,

whose construction will be explained in the next

section.

3. Homotopy If f : N ! M and g : N ! M are

homotopic maps, then the pullback maps f

= g

are equal. If in addition, f and g are K-oriented,

proper maps which are homotopic via proper

maps, then the Gysin maps f

= g

are equal.

4. Excision Let M

be a closed subset of M and U

be an open subset of M such that U is contained

in the interior of M

. Then the inclusion of pairs

(MnU, M

nU) ,!(M, M

) induces an isomorph-

ism in K-theory, K



(M, M

) ﬃ K



(MnU, M

nU).

5. Exactness Let M

be a closed subset of M. Then

there is a six-term exact sequence in K-theory,

ðM; M

Þ!K

ðMÞ! K

ðM

"#

ðM

Þ K

ðMÞ K

ðM; M

6. Cup product There is a canonical map given by

external tensor prod uct, K

(M)  K

(N) !

iþj

(M  N). When N = M, one can compose this

with the homomorphism induced by the diagonal

map M ! M  M given by x ! (x, x), to get a cup

product, K

(M)  K

(M) ! K

pþq

(M).

7. Bott periodicity This is arguably the most impor-

tant property of K-theory. It says that the zero-

section embedding 

: M ,!M  R

induces a

Gysin isomorphism, 

: K



(M)!

ﬃ

þn

(M  R

which is given as follows. Let 

: M  R

! M

and 

: M  R

! R

denote the projections

onto the factors, and b = 

1 2 K

) the Bott

element, where  : {0} ,!R

is the inclusion of the

origin. Then the Bott periodicity isomorphism is

given by 

(x) = 

(x) [ 

(b) 2 K

þn

(M  R

)

for all x 2 K



(M).

Using the fact that any v ector bundle over a

contractible space is trivial, together with Bott’s

periodicity theorem, one deduces the calculation

of the K-theory of spheres. The calculation for the

odd-dimensional spheres given, K

2n1

) ﬃ Z ﬃ

2n1

), and for the even-dimensional spheres

2n1

) ﬃ Z

and K

) ﬃ {0}, for all n 1.

There is a natural homomorphism of rings called

the Chern character, Ch : K



(X) ! H



(X, Q) which

is characterized by the following axioms:

1. Naturality If f : N ! M is a smooth map, and if

E is a vector bundle over M, then Ch(f

(E)) =



(Ch(E)).

2. Additivity Ch(E  F) = Ch(E) þ Ch(F).

3. Normalization If L is the canonical line bundle

over CP

which restricts to the Hopf line bundle

over CP

, then Ch(L) = exp (x), where x is the

generator of H

(CP

, Z) ﬃ Z.

Atiyah and Hirzebruch, cf. Karoubi (1978), also

proved that the Chern character induces an iso-

morphism of the rings K



(X)  Q and H



(X, Q). The

Chern–Weil representative of the Chern character is

tr(exp((i=2)

)), where 

is the curvature of a

Hermitian connection on E.

There are many variants of K-theor y, such as

KO-theory, where the unitary group is replaced

by the orthogonal group, which is periodic of

order eight, and G-equivariant K-theory, where G

is a compact Lie group. K-theory and its variants

have many interesting applications such as deter-

mining the maximum number of linearly inde-

pendent vector fields on spheres, which is due to

Adams, cf. Karoubi (1978). We will content

ourselves with the description of two important

applications.

Grothendieck–Riemann–Roch Theorem

for Smooth Manifolds

Recall that an oriented real vector bundle E over M is

said to be a spin

vector bundle if the bundle of

oriented frames on E, SO(E) has a circle bundle

Spin

(E) such that the restriction to each fiber yields

the central extension 0 ! U(1) ! Spin

(n) !

SO(n) ! 0 that defines the group Spin

(n), where n

is the rank of E. It turns out that the obstruction to the

existence of a spin

structure on E is the third integral

Stieffel–Whitney class of E, W

(E) 2 H

(M, Z).

K-Theory 247

A generalization of Bott periodicity is the Thom

isomorphism in K-theory. It says that if  : E ! M is

arank-n spin

vector bundle over M, then the zero-

section embedding 

: M ,!E induces a Gysin iso-

morphism, 

: K



(M) ﬃ K

þn

(E), which is given as

follows. There is a canonical element 

1 2 K

(E)

called the Thom class in K-theory, which is character-

ized by the property that 

1 restricts to give the Bott

class on each fiber. Then the Thom isomorphism in K-

theory is given by 

(x) = 

(x) [ 

1 2 K

þn

(E)for

all x 2 K



(M). For canonical representatives of the

Thom class, cf. Mathai–Quillen Formalism, or Mathai

and Quillen (1986).

Recall the definition of the Gysin map for smooth

embeddings. Let X be a smooth, compact manifold,

and Y a smooth manifold. Let h : X ! Y be a smooth

embedding that is K-oriented. Since TX  TX has a

canonical almost-complex structure, it follows that

the normal bundle N

X = h



(TY)=TX is a spinC

vector bundle. If 

: X ,!N

X is the zero-section

embedding, then we have the Thom isomorphism



: K



(X) !

ﬃ

þn

X), where n = dim(Y) dim(X)

is the codimension of the embedding. Upon choosing a

Riemannian metric on Y, there is a diffeomorphism 

from a tubular neighborhood U of h(X)ontoa

neighborhood of the zero section in the normal bundle

(X). That is, 

: K



X) !

ﬃ



(U). For any open

subset j : U ,!Y, the extension by zero defines a

homomorphism j : K



(U) ! K



(Y). Then the Gysin

map of the embedding h is defined as h

= j  





: K



(X) ! K

þn

(Y), which turns out to be inde-

pendent of the choices made.

Next recall the definition of the Gysin map for

smooth submersions. Let  : Y ! Z be a smooth

submersion of smooth manifolds, which is K-

oriented and a proper map. Since every smooth

compact manifold can be smoothly embedded in

for q sufficiently large, a parametrized version

yields an embedding  : Y ,!Z  R

that is spinC.

Therefore the Gysin map is a homomorphism





(Y) !K

þa

(Z R

), where a= dim(Z) þ2q

dim(Y). Let 

:Z ,!Z R

denote the zero-section

embedding. Then we have the Thom isomo rphism





(Z) !

ﬃ

þ2q

(Z R

). Then the Gysin map

of the submersion  is defined as 

= 

(

)

1



(Y) ! K

þb

(Z), where b = dim( Y) dim(Z), and

turns out to be independent of the choices made.

Let f : N ! M be a smooth proper map that is

K-oriented. Then f can be canonically factored, first

into the smooth embedding gr(f ):N ,!N  M,

which is the graph of the function, that is,

gr(f )(x ) = (x, f (x)), and which is K-oriented. The

Gysin map is gr(f )

: K



(N) ! K

þdim(M)

(N  M).

Second, the projection p

: N  M ! M is a

K-oriented proper submersion, when restricted to

the image of gr(f). The Gys in map is p

: K



(M 

N) ! K

þb

(M), where b = dim(N). The Gysin map

of f is defined as f

= p

 gr(f )

: K



(N) ! K

þd

(M),

where d = dim(M) þ dim(N).

Given such a smooth proper map f : N ! M that

is K-oriented. Then there are Gysin maps in

cohomology, f



: H



(N, Q) ! H

þd

(M, Q) (where

we consider the Z

-grading given by even and odd

degree), and in K-theory, f

: K



(N) ! K

þd

(M)

which increases the degree by d = dim(M) þ

dim(N). The Groth endieck–Riemann–Roch theorem

due to Atiyah and Hirzebruch, cf. Karoubi 1978,in

the smooth category can be phrased as the commu-

tativity of the diagram,



ðNÞ !

þd

ðMÞ

ToddðTNÞ[Ch

ToddðTMÞ[Ch



ðN; QÞ !



þd

ðM; QÞ

That is,

Chðf

ðÞÞ [ ToddðTMÞ¼f



ðChð Þ[ToddðTNÞÞ

for all  2 K



(N), where Todd(E) is the Todd genus

characteristic class of a Hermitian vector bundle E

over M. The Chern–Weil representative of the Todd

genus is

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det

ði=2Þ

tanh ði=2Þ

ðÞ



where 

is the curvature of a Hermitian connection

on E. There are many useful vari ants of this

beautiful formula.

The Atiyah–Singer Index Theorem

The 2004 Abel Prize citation mentions the Atiyah–

Singer (1971) index theore m as being one of the

greatest achievements of twentieth-century mathe-

matics. It has stimulated considerable interaction

between mathematicians and mathematical physi-

cists. We content ourselves here with a rudimentary

description of the results.

Let F be the space of all Fredholm operators on

an infinite-dimensional complex Hilbert space H.

Recall that an operator A is said to be Fredholm if

both the kernel and cokernel of A are finite

dimensional. The index of such a Fredholm operator

is index(A) = dim(ker(A))  dim(coker(A)) 2 Z. The

index map is continuous, so it induces a map on the

connected components of F, which turns out to be

an isomorphism.

248 K-Theory

K-theory is naturally related to the space of all

Fredholm operators F endowed with the norm

topology. Any continuous map A : X !F from a

compact space to F has an index in K

(X), which

is given by index(A) = ker (A)  coker(A) in the

special case when dim(ker(A))(x) is constant in x 2

X. In general, one uses the fact that the index is

stable under compact perturbation, and shows that

one can always achieve the special case after a

compact perturbation. It is again the case that the

index map is continuous, and so induces a map,

index : [X, F] ! K

(X), which turns out to be an

isomorphism, thanks to a fundamental theorem

of Kuiper which proves that the group of all

invertible operators on an infini te-dimensional

complex Hilbert space is con tractible in the norm

topology.

Now let  : N ! Z be a fiber bundle with typical

fiber a smooth compact manifold M, where N and Z

are also smooth compac t manifolds. Consider a

smooth family of elliptic operators D = { D

}

z2Z

along the fibers of , parametrized by Z, where

: C

(

1

(z), E j



1

(z)

) ! C

(

1

(z), F j



1

(z)

) and

E, F are vector bundles over N. Such a family of

elliptic operators has a symbol

ðDÞ : 



ðEÞ!



ðFÞ

where  : T



(N=Z) ! N is the projection and



(N=Z) is the vertical cotangent bundle. Ellipticity

for the family is the condition that (D)isan

isomorphism outside the zero section, so that the

triple (



(E), 



(F), (D)) determines an element in



(N=Z)) denoted by (D).

The analytic index of the family D is index(D) 2

(Z), and it turns out that it only depends on the

class of the symbol (D) 2 K



(N=Z)), so the

analytic index can be viewed as a homomorphism,

index : K

ðT



ðN=ZÞÞ ! K

ðZÞ

Consider an embedding  : N ,!Z  R

that is

compatible with the projection  : N ! Z. The

fiberwise differential is an embedding d : T(N=Z) !

Z  R

, which induces a Gysin map

d

: K

ðTðN=ZÞÞ ! K

ðZ  R

upon identifying T



(N=Z)withT(N=Z). Let

j : Z ! Z  R

be the inclusion j(z) = (z, 0). It

induces the Bott isomorphism j

: K

(Z) ﬃ K

(Z 

). The topological index of the family D is, by

definition,

index

¼ j

1

 d

: K

ðT



ðN=ZÞÞ ! K

ðZÞ

The Atiyah–Singer (1971) index theorem

for families of elliptic operators D asserts the

equality of the analytic index and the topological

index,

indexðDÞ¼index

ððDÞÞ 2 K

ðZÞ

Combined with the Grothendieck–Riemann–Roch

theorem, one has the following exquisite formula in



(Z, Q):

ChðindexðDÞÞ ¼ 







fToddðT



ðN=ZÞÞ [ ChððDÞÞg

where  : T



(N=Z) ! N is the projection.

The map sending a complex vector bundle E over

Z to its determinant line bundle det(E) =

max

induces a homomorphism, det : K

(Z) ! 

(Pic(Z)),

where 

(Pic(Z)) denotes the isomorphism classes of

complex line bundles over Z. Then

ðdetðindexðDÞÞÞ

¼ 







fToddðT



ðN=ZÞÞ [ ChððDÞÞg



½2

where

[2]

denotes the degree-2 component, and the

left-hand side denotes the first Chern class of the

determinant line bundle of the index class. This

formula is often used in the study of anomalies in

physics.

K-Theory of C



-Algebras

The Gelfand–Naimark theorem asser ts that unital

abelian C



-algebras A can be identified with the

space of continuous functions C(X), where X is the

compact Hausdorff space known as the spectrum of

A, consisting of characters of A. Conve rsely, given a

compact Hausdorff space X, the characters of C(X)

consist of the evaluation maps at points of X.

Let E be a vector bundle over X. Then there is a

vector bundle F over X such that E  F ﬃ X  C

Setting A = C(X ), M= C(X, E), N = C(X, F), we

see that MN ﬃA

, showing that each vector

bundle E over X determines a canonical finite

projective module M over A. The converse is also

true and is a result of Serre and Swan, cf. Blackadar

(1986), which asserts that every finite projective

module M over A is the space of all continuous

sections of a vector bundle over X. So we have an

equivalence of the category of vector bundles over X

and the category of finite projective modules over A.

This motivates the following generalization of

topological K-theory for a general unital C



-algebra

A. Let Proj(A) denote the isomorphism classes of

finite projective modules over A. It is a commutative

semigroup under the ope ration of direct sum, which

can be made into the commutative group K

(A)as

follows: K

(A) is generated by pairs ([M], [N]),

together with the relation ([M], [N]) = ([M

], [N

])

if MN

GﬃM

NG for some [G] 2 Proj

K-Theory 249

(A). Also K

(A) = 

(GL(1, A)) where GL(1, A)

denotes the direct limit of GL(n, A) where

ðGLðn; AÞ embeds in GLðn þ 1; AÞ as 1  GLðn; AÞ:

Then, defining K

(A) = 

j1

(GL(1, A)) for j 1,

together with generalized Bott periodicity which

asserts that there is a canonical isomorphism



j1

(GL(1, A)) ﬃ 

jþ1

(GL (1, A)), we see that



(A) = K

(A)  K

(A) is a generalized periodic

cohomology theory. If A is a C



-algebra without

unit, then consider A

= A  C, with product given

by (a, )(b, ) = (ab þ a þ b, ) with unit (0, 1).

The projection p : A

!C defined as p(a, ) = 

induces a map p

: K



) ! K



(C). In the nonunital

case, K



(A) is defined as ker(p

). Observe that

(A) = K

), but this is often not the case with

. It is easy to see that when A has a unit, then the

two definitions of K

agree. An important caveat in

the case of noncommutative C



-algebras is that the

K-theory is often not a ring as there is no analog of

the tensor product operation.

Some of the basic properties of K-theory are listed

as follows. Details can be found in Blackadar

(1986).

1. Cup product A continuous bilinear map of



-algebras, A  B ! C, induces a cup product,

(A)  K

(B) ! K

iþj

(C).

In particular, the continuous product A  A !A

induces a cup product homomorphis m,

(A)  K

(A) ! K

iþj

(A).

2. Induced homomorphism If f : A ! B is a homo-

morphism of C



-algebras, then there is an

induced homomorphism, f

: K



(A) ! K



(B).

3. Homotopy If f : A ! B and g : A ! B are

homomorphisms of C



-algebras that are homo-

topic, the induced homomorphisms on K-theory



= g



are equal.

4. Excision If I is a closed two-sided ideal in A,

then there is a six- term exact sequence in

K-theory,

ðIÞ!K

ðAÞ!K

ðA=IÞ

"#

ðA=IÞ K

ðAÞ  K

ðIÞ

5. Morita invariance The inclusion homomorph-

ism of A into the top left of the diagonal in

(A) induces an isomorphism in K-theory,



(A) ﬃ K



(A)).

6. Continuity Let A = lim

n !1

be a C



-direct

limit. Then, K



(A) = lim

n !1



7. Stability Let K be a C



-algebra of all compact

operators on an infinite-dimensional complex

Hilbert space. Then since K= lim

n !1

(C)is

a C



-direct limit, we see that K



(A K) =

lim

n !1



(A  M

(C)) = K



(A).

8. Bott periodicity The continuous product A 

C !A induces the cup product K

(A) K

iþj

(A). The computation by Bott asserts

that there is a canonical element b 2K

gives an isomorphism K

Bott periodicity asserts that the cup product

with b gives rise to an isom orphism K

(A) ﬃ

iþ2j

(A).

We mention in passing that Connes has defined a

Chern character homomorphism, Ch : K



(A) !



(A), mapping into the entire cyclic homology

of A, having similar properties as the ordinary

Chern character. Due to space constraints, it will

not be defined here.

A C



-Algebra Generalization of the

Atiyah–Singer Index Theorem and

the Baum–Connes Conjecture

We content ourselves here with a rudimentary

account of the C



-algebra generalization of the

Atiyah–Singer index theorem and the Baum–Connes

conjecture, and its relevance to the quantum Hall

effect and strict deformation quantization. Let A be

a C



-algebra.

Let H

= A H, which is the analog of a Hilbert

space. Let F

be the space of all A-Fredh olm

operators on H

. Recall that an operator T is said to

be A-Fredholm if both the kernel and cokernel of T þ

K are closed and finitely generated projective modules,

where K is an A-compact operator. The space of

A-compact operators is by definition the closure of

the A-finite rank operators. The index of T is

indexðTÞ¼½kerðT þ KÞ  ½cokerðT þ KÞ 2 K

ðAÞ

The index map turns out to be well defined and

independent of the choice of A-compact perturba-

tion K. It is continuous, so it induces a map on the

connected components of F

, which turns out to

be an isomorphism, by a theorem of Mingo

(cf. Rosenberg (1983, 1989)).

Now let M be a smooth compact manifold. An

A-vector bundle over M is a locally trivial Banach

vector bundle E over M whose fibers have the

structure of finitely generated left A-modules, with

morphisms respecting the A-module structure. The

isomorphism classes of A-vector bundles over M

form a commutative semigroup under direct sums,

and the associated commutative group is easily

identified with K

(C(M)  A). Let D : C

(M, E) !

(M, F) be an elliptic A-operator acting between

smooth sections of A-vector bundles E, F over M.It

250 K-Theory

turns out that by elliptic regularity, such an operator

is A-Fredholm, and has an analytic index,

indexðDÞ2K

ðAÞ

Associated to each such operat or is a symbol

ðDÞ : 



ðEÞ!



ðFÞ

where  : T



M ! M is the projection. Ellipticity is

the condition that (D) is an isomorphism outside

the zero section, so that the triple (



(E), 



(F ), (D))

determines an element in K



M)  A) denoted

by (D). It turns out that the analytic index of D

depends only on the class (D) 2 K



M)  A).

Therefore, the analytic index can be viewed as a

homomorphism,

index : K

ðC

ðT



MÞAÞ!K

ðAÞ

Consider an embedding  : M ,!R

, which induces

an embedding d : TM ! R

. The associated Gysin

map is d

: K



M)  A) ! K

)  A).

Let j : {0} ! R

denote inclusion of the origin in R

It induces a Gysin map j

: K

(A) ! K

)  A)

which is the Bott periodicity isomorphism. Then the

topological index is the homomorphism

index

¼ j

1

 d

: K

ðC

ðT



MÞAÞ!K

ðAÞ

The C



-generalization of the Atiyah–Singer

index theorem due to Mishchenko–Formenko, cf.

Kasparov (1988), asserts the equality of the

analytic index and the topological index,

indexðDÞ¼index

ððDÞÞ 2 K

ðAÞ

Now let M be a compact even-dimensional

spin

manifold. Then there is a spin

Dirac

operator D : C

(M, S

) ! C

(M, S



), where S



the bundle of half-spinors on T



M  L,whereL is

a line bundle over M with the property that the

first Chern class of L modulo 2, c

(L)mo d 2 is

equal to the second Stieffel–Whitney class of M,

(M). Let  be a torsion-free discrete group, and

B be its c lassifying space. It is a paracom pact

space with the property that it is the quotient of 

acting freely on a contractible space E.LetC



()

denote the reduced group C



-algebra, and consider

the canonical flat C



() bundle V over B defined

as follows:

V¼fE  C



ðÞg=

where  acts on the left on C



() and on the right on

E. Let f : M ! B be a contin uous map. Then f



is a flat C



()-bundle over M. Upon choosing a flat

connection on f



V, we can couple the spin

Dirac

operator D

to act on sections of S



 f



V. The

ellipticity of D

ensures that it is a C



()-Fredholm

operator, so it has an an alytic index, index(D

) 2



()) by the earlier discussion, which is

also equal to the topological index index

((D

)) 2



()).

By Ba um, Connes, and Douglas, the K-homology

of B, K

(B), is generated by the triples (M, E, f )as

described above, modulo relations that we will not

present here because of space constraints. The

assembly map

 : K

ðBÞ!K

ðC



ðÞÞ

is a homom orphism given by ([(M, E, f )]) =

index(D

). The Baum–Connes conjecture asserts

that  is an isomorphism. There are variants of

this conjecture when  has torsion. The Baum–

Connes conjecture has been verified when  is an

amenable group or, for instance, a word hyperbolic

group. There are also variants of this conjecture for

certain foliations and groupoids, and is an extremely

active area of research. The injectivity of the

assembly map is related to the Novikov conjecture

on the homotopy invariance of the higher signatures

(Kasparov 1988), and the obstructions to the

existence of Riemannian metrics of positive scalar

curvature on compact spin manifolds (Rosenberg

1983, 1989). A variant of the Baum–Connes

conjecture, where the reduced group C



-algebra is

replaced by the twisted reduced group C



-algebra, is

used in the analysis of the noncommutative geome-

try approach to the integer and fractional quantum

Hall effect, and also the gaps in the spectrum of

magnetic Schro¨ dinger operators (Bellissard et al.

1994, Marcolli and Mathai 2001).

Twisted K-theory and the Chern

Character

We begin by reviewing some results due to Dixmier

and Douady (1963).LetM be a smooth manifold, let

H denote an infinite-dimensional, separable, Hilbert

space and let K be the C



-algebra of compact

operators on H.LetU(H) denote the group of

unitary operators on H endowed with the strong

operator topology and let PU(H) = U(H)=U(1) be the

projective unitary group with the quotient space

topology, where U(1) consists of scalar multiples of

the identity operator on H of norm equal to 1. Since

U(H) is contractible in the operator norm topology, it

follows that PU(H) = BU(1) is an Eilenberg–MacLane

space K(Z , 2). Therefore, BPU(H) is an Eilenberg–

MacLane space K(Z, 3). That is, principal PU(H)

bundles P over X are classified up to isomorphism by

K-Theory 251

the Dixmier–Douady class DD(P)inH

(X, Z)and

conversely.

For g 2 U(H), let Ad(g) deno te the automorphism

T !gTg

1

of K. As is well known, Ad is a

continuous homomorphism of U(H), given the

strong operator topology, onto Aut(K) with kernel

the circle of scalar multiples of the identity where

Aut(K) is given the point-norm topology. Under this

homomorphism we may identify PU(H) with

Aut(K). Define an Azumaya bundle to be a locally

trivial bundle E over X with fiber K and structure

group Aut(K). They are of the form K

= {P K}=

PU(H) and isomorphism classes of Azumaya bundles

are also parametrized by their Dixmier–Douady

class DD(P)inH

(X, Z) and conversely.

Since KKﬃK, the isomorphism classes of

locally trivial bundles over X with fiber K and

structure group Aut(K) form a group under the

tensor product, where the inverse of such a bundle

is the conjugate bundle. This group is known as

the infinite Brauer group and is denoted by Br

(X).

So, a restatement of the Dixmier–Douady theorem

is that Br

(X) ﬃ H

(X, Z).H

(X, Z) can also

be described in terms of bundle gerbes (Murray

1996).

The twisted K-theory, K



(X, P ), is defined as the

K-theory of the C



-algebra of continuous sections of

the Azumaya bundle K

, K



(C(X, K

)). It was

studied in the torsion case by Donovan and Karoubi,

where one can replace the compact operators K by

finite-dimensional matrices, and was studied in the

general case by Rosenb erg (1983, 1989). Let F be

the space of all Fredholm operators endowed with

the norm topology. Then, one can form the bundle

of Fredholm operators F

= {P F}=PU(H), where

PU(H) acts on F via the a djoint action. Consider the

fibration K

! GL(C

), where C

= {P C}=

PU(H) and C= B(H)=K is the Calkin algebra. Since



(C(X, K

)) = {0}, we see that 

(C(X, F

)) =



(C(X, GL(C

))). Consider the short exact sequence

of C



-algebras,

0 ! CðX; K

Þ!CðX; B

Þ!CðX; C

Þ!0

where B

= {P B(H)}=PU(H) and where PU(H)

acts on B(H) via the adjoint action. It gives rise to

a six-term exact sequence

ðCðX;K

ÞÞ ! K

ðCðX;B

ÞÞ ! K

ðCðX;C

ÞÞ

index"#

ðCðX;C

ÞÞ  K

ðCðX;B

ÞÞ  K

ðCðX;K

ÞÞ

By definition, K

(C(X,C

)) ﬃ

(C(X,GL(1,C

)))

and a standard argument shows that this is also

equal to 

(C(X,GL(C

))). By Kuiper’s theorem, it is

not difficult to see that K



(C(X,B

))={0}.

Therefore,

index: 

ðCðX; F

ÞÞ ! K

ðX; PÞ

is an isomorphism. Let X

be a closed subset of X,

and I

be the closed ideal of sections of K

that

vanish on X

. Then K



(X, X

, P) is by definition



). A geometric description of twisted K-the ory

in terms of modules for bundle gerbes is described in

Bouwknegt et al. (2002).

Some of the basic propert ies of twisted K-theory

are listed as follows. Many of these properties

follow from the corresponding properties for the

K-theory of C



-algebras. See Atiyah and Segal and

Bouwknegt et al. (2002).

1. Normalization If P is trivial, then K



(M, P) =



(M).

2. Module property K



(M, P) is a module over

(M).

3. Pullback If f : N ! M is a continuous map,

and P a principal PU(H) bundle over M, then

there is a pullback homomorphism f : K



(M, P) !



(N, f(P)).

4. Push-forwa rd Let f : N !M be a smooth proper

map between compact manifolds which is K-

oriented, that is, TN  f



TM is a spin

vector

bundle over N.LetP be a principal PU(H) bundle

over M. Then there is a pushforward homomorph-

ism, also called a Gysin map, f : K



(N, f

(P)) !

þd

(M, P), where d = dim M dim N.

5. Homotopy If f : N ! M and g : N ! M are

homotopic maps, then the pullback maps f

= g

are equal. If in addition, f and g are K-oriented,

then the pushforward maps f

= g

are equal.

6. Excision Let M

be a closed subset of M and U

be an open subset of M such that U is contained

in the interior of M

. Then the inclusion of

pairs (MnU, M

nU) ,!(M, M

) induces an iso-

morphism in K-theory, K



(M, M

, P) ﬃ



(MnU, M

nU, P j

MnU

7. Exactness Let M

be a closed subset of M and

 : M

! M be the inclusion. Let P be a principal

PU(H) bundle over M. Then the short exact

sequence

0 ! I

! CðM; K

Þ!CM

; K



! 0

gives rise to the six-term exact sequence in K-theory,

ðM; M

; PÞ!K

ðM; PÞ!K

ðM

;

ðPÞÞ

"#

ðM

;

ðPÞÞ K

ðM; PÞ K

ðM; M

; PÞ

252 K-Theory

8. Cup product Let P be a principal PU(H) bundle

over M and Q be a principal PU(H) bundle over N.

An identification HHﬃHgives rise to a principal

PU(H) bundle P  Q over M  N whose Dixmier–

Douady invariant is DD(P  Q) = p



(DD(P)) þ



(DD(Q)), where p

denote projections onto the

jth factor, j = 1, 2. Then there is a canonical map

given by external tensor product,

ðM; PÞK

ðN; QÞ!K

iþj

ðM  N; P  QÞ

called the cup product.

9. Bott periodicity Let P be a principal PU(H)

bundle over M. Bott periodicity says that there is

a canonical isomorphism



ðM; PÞﬃK

þn

ðM  R

;ðPÞÞ

where  : M  R

! M is the projection onto the

first factor. Let b 2 K

) be the Bott element.

Then the isomorphism above is given by 

(x) [

b 2 K

þn

(M  R

, 

(P)) for all x 2 K



(M, P).

There is a natural homomorphism of rings called the

twisted Chern character, which depends both on a

choice of P and a de Rham representative H of DD(P),

: K



ðM; PÞ!H



ðM; HÞ

Here H



(M, H) denotes the twisted cohomology,

which is by definition the cohomology of the

complex (



(M), d  H^). The twisted Chern char-

acter is characterized by the following axioms:

1. Naturality If f : N ! M is a smooth map, and if

x 2 K



(M, P), then Ch

f(P)

(x)) = f



(Ch

(x)).

2. Additivity If x, y 2 K



(M, P), then Ch

(x  y) =

(x) þ Ch

(y).

3. Ch

respects the K

(M)-module structure of

(M, P).

4. Normalization If P is trivial, then Ch

reduces

to the ordinary Chern character Ch.

It turns out that the twisted Chern character

induces an isomorphism of the rings K



(M, P)  Q

and H



(M, H). The Chern–Weil representative of the

twisted Chern character is derived in Bouwknegt

et al. (2002).

Twisted K-Theory and Duality in Type II

String Theories

Let E be an oriented S

-bundle over M,

! E



characterized by its first Chern class c

(E) 2

(M, Z), in the presence of (possibly nontrivial)

H-flux H 2 H

(E, Z). We will argue that the T-dual

of E is again an oriented S

-bundle over M, denoted

!

^

supporting H-flux

H 2 H

(

E, Z), such that

EÞ¼



H; c

ðEÞ¼^



where 



: H

(E, Z) !H

k1

(M, Z) and, similarly, 



denote the pushforward maps. Then we can form

the following commutative diagr am:

The correspondence space E 

E is a circle bundle

over E with first Chern class 



(

E)), and it is also

a circle bundle over

E with first Chern class





(E)), by the commutativity of the diagram

above. If

E = E or if

E = M  S

, then the correspon-

dence space E 

E is diffeomorphic to E  S

T-duality gives an isomorphism of the twisted

K-theories of E and

E as well as an isomorphism

between the twisted cohomologies of E and

E, and

can be expressed in the following commutative

diagram:



ðE; P Þ!

þ1

PÞ

##Ch



ðE; HÞ!



þ1

HÞ

where the horizontal arrows are isomorphisms. Here

P is a principal PU(H) bundle over E such that

DD(P) = H and

P is a principal PU(H) bundle over

E such that DD(

P) = H. We refer to Bouwknegt

et al. (2004) for details. The T-duality isomorphism

above gives compelling evidence that a type IIA

string theory A on a circle bundle of radius R in the

presence of a background H -flux, and a type IIB

string theory B on a ‘‘T-dual’’ circle bundle of radius

E ×

K-Theory 253

1/R in the presence of a ‘‘T-dual’’ background H-flux,

are equivalent in the sense that the string states of

string theory A are in canonical one-to-one correspon-

dence with the string states of string theory B.

We briefly m ention two other applications of

twisted K-theory. Consider the adjoint action of a

compact connected simple Lie group G on itself,

and the corresponding twisted G-equivariant

K-theory, twisted by a multiple of the generator

of H

(G, Z). The relevance of the equivariant case

to conformal field theory was highlighted by the

result of Freed, Hopkins and Teleman (see Freed

(2002)) that it is graded isomorphic to the

Verlinde algebra of G, with a shift given by the

dual Coxeter number. Here the Verlinde algebra

consists of equivalence classes of positive-energy

representations of the l oop group of G which was

originally shown to be a ring in a rather nontrivial

way. On the other hand, the ring structure of the

twisted G-equivariant K-theory of G is ju st

induced by the product on G, which makes this

result all the more remarkable.

Fractional analytic index theory, developed in

Mathai et al. is a generalization of Atiyah–Singer

index theory, assigning a fractional-valued analytic

index to each projective elliptic operator on a compact

manifold, where the fraction need not be an integer.

These projective elliptic operators act on projective

vector bundles, where the usual compatibility condi-

tion on triple overlaps to give a global vector bundle,

may fail by a scalar factor. These are the geometric

objects in twisted K-theory, when the twist is torsion.

In Mathai et al., a fractional index theorem is

proved, computing the fractional-valued analytic

index of projective elliptic operators essentially in

terms of topological data. The Dirac operator in

the absence of a spin structure is also defined there

for the first time resolving a long standing mystery,

and its index is computed.

Some topics not covered in this brief account of

K-theory include: KK-theory, cf. Blackadar (1986)

and Kasparov (1988), which is natural setting for

the Atiyah–Singer index theorem and its general-

izations, as well as higher algebraic K-theory.

See also: C*-Algebras and Their Classification;

Characteristic Classes; Cohomology Theories;

Equivariant Cohomology and the Cartan Model; Gerbes

in Quantum Field Theory; Index Theorems; Intersection

Theory; Mathai–Quillen Formalism; Spectral Sequences.

Further Reading

Atiyah MF and Segal G Twisted K-theory, math.KT/0407054.

Atiyah MF and Singer IM (1971) The index of elliptic operators,

IV. Annals of Mathematics 93: 119–138.

Bellissard J, van Elst A, and Schulz-Baldes H (1994) The

noncommutative geometry of the quantum Hall effect. Journal

of Mathematical Physics 35: 5373–5451.

Blackadar B (1986) K-Theory for Operator Algebras. In:

Mathematical Sciences Research Institute Publications,

vol. 5, viiiþ338 pp. New York: Springer.

Bouwknegt P, Carey A, Mathai V, Murray MK, and Stevenson D

(2002) Twisted K -theory and the K-theory of bundle gerbes.

Communications in Mathematical Physics 228(1): 17–49.

Bouwknegt P, Evslin J, and Mathai V (2004) T-duality: topology

change from H-flux. Communications in Mathematical Phy-

sics 249: 383 (hep-th/0306062).

Bouwknegt P, Evslin J, and Mathai V (2004b) On the topology and

flux of T-dual manifolds. Physical Review Letters 92: 181601.

Dixmier J and Douady A (1963) Champs continues d’espaces

hilbertiens at de C



-alge`bres.Bull.Soc.Math.France91: 227–284.

Freed DS (2002) Twisted K-theory and loop groups. Proceedings

of the International Congress of Mathematicians (Beijing,

2002) vol. III pp. 419–430.

Karoubi M (1978) K-theory. An Introduction Grundlehren der

Mathematischen Wissenschaften, Band 226, xviiiþ308 pp.

Berlin: Springer.

Kasparov GG (1988) Equivariant KK-theory and the Novikov

conjecture. Invent. Math. 91(1): 147–201.

Marcolli M and Mathai V (2001) Twisted index theory on good

orbifolds, II: fractional quantum numbers. Communications in

Mathematical Physics 217(1): 55–87.

Mathai V, Melrose RB, and Singer IM, The fractional analytic

index, math.DG/0402329.

Mathai V and Quillen DG (1986) Superconnections, Thom classes

and equivariant differential forms. Topology 26: 85–110.

Murray MK (1996) Bundle gerbes. J. London Math. Soc. 54:

403–416.

Rosenberg J (1983) C



-algebras, positive scalar curvature, and the

Novikov conjecture. Inst. Hautes tudes Sci. Publ. Math 58:

197–212.

Rosenberg J (1989) Continuous-trace algebras from the bundle

theoretic point of view. J. Austral. Math. Soc. Ser. A 47(3):

368–381.

254 K-Theory

Lagrangian Dispersion (Passive Scalar)

G Falkovich, Weizmann Institute of Science, Rehovot,

Israel

Introduction

To describe transport by a random flow, one needs

to apply the statistical methods to the motion of

fluid particles, that is, to the Lagrangian dynamics.

We first present the propagators describing evolving

probability distributions of different configurations

of fluid particles. We then use those propagators to

describe decay and steady states of a passive scalar

field transported by random flows.

Consider an evolut ion of a passive scalar tracer

(r, t) in a random flow. The mean value of the

scalar tracer at a given point is an average over

values brought by different trajectories:

ðr; sÞ

Pðr; s; R; 0ÞðR; 0ÞdR ½1

Here, P(r, s; R, t) is the probability density function

(PDF) to find the particle at time t at position R

given its position r at time s. That PDF is called the

propagator or the Green function. Multipoint

correlation functions of the tracer

ðr;sÞ ðr

;s Þ...ðr

;sÞ

ðr;s; R; 0ÞðR

;0Þ...ðR

;0ÞdR ½2

are expressed via the multiparticle Green functions

which are the joint PDFs of the equal-time

positions

R= (R

,...,R

)ofN fluid trajectories.

The trajectory of the fluid particle that passes at

time s through the point r is described by the vector

R(t; r, s) which satisfies R(t; r, t) = r and the stochas-

tic equation

R ¼ vðR; tÞþuðtÞ½3

Here, u(t) describes the molecular Brownian motion

with zero average and covariance hu

(t) u

= 2

(t  t

). We also consider macroscopic

velocity v as random with various statistical properties

in space and time. There is a clear scale separation

between macroscopic velocity v and molecular

diffusion u that allows one to treat them separately.

Using [3], one can write the Green’s function as

an integral over paths that satisfy q(s) = r an d

q(t) = R:

Pðr; s; R; tÞ¼



DpDq exp 

{pðÞ½

qðÞ



 vðqðÞ;ÞuðÞd



v;u

½4



DpDq exp 

½{pðÞð

qðÞ



 vðqðÞ;ÞÞ þ p

ðÞd



½5



Dq exp





4

qðÞ

 vðqðÞ;Þ

d



¼ Pðr; s; R; tjvÞ

½6

The integration over the auxiliary field p in [4]

enforces the delta function of [3]. One passes from

[4] to [5] by averaging over the Gaussian Brownian

noise, and from [5] to [6] by calcu lating Gaussian

integral over p.

Generally, exact calculations are only possible for

Gaussian random proces ses short-correlated in time-

like in [5]. The simplest case is the Brownian motion

when the advect ion is absent. One then obtains from

[6] the Gaussian PDF of the displacement:

PðR; tÞ¼ð4tÞ

d=2

R

=ð4tÞ

½7

which satisfies the heat equation (@

 r

) P(r, t) = 0.

The short-correlated case is far from being an exotic

exception but rather presents a long-time limit of an

integral of any finite-correlated random function.

Indeed, such an integral can be presented as a sum of

many independent equally distributed random numbers,

the statistics of such sums is a subject of the central limit

theorem. One can move beyond the central limit

theorem considering the correlation time finite (yet

small comparing to the time of evolution). Such

generalization is the subject of the large deviation

theory. Consider some quantity X whichisanintegral

of some random function over time t much larger than

the correlation time .Att  , X behaves as a sum of

many independent identically distributed random

numbers y

: X =

with N / t=. The generating

function he

i of the moments of X is the product,

i= e

NS(z)

, where we have denoted he

ie

S(z)

(assuming that the generatin g function he

i exists for

all complex z). The PDF P(X) is given by the inverse

Laplace transform (2i)

1

zXþNS(z)

dz with the

integral over any axis parallel to the imaginary one.

For X / N, the integral is dominated by the saddle point

such that S

) = X=N and

PðXÞ/e

NHðX=NhyiÞ

½8

Here H = S(z

) þ z

) is the function of the

variable X=N hyi; it is called entropy function as it

appears also in the thermodynamic limit in statis-

tical physics. A few important properties of H (also

called rate or Crame´r function) may be established

independently of the distribution P(y). It is a convex

function which takes its minimum at zero, that is,

for X equal to the mean value hXi= NS

(0). The

minimal value of H vanishes since S(0) = 0. The

entropy is quadratic around its minimum with

(0) =

1

, where =S

(0) is the variance of y.

We thus see that the mean value hXi= Nhyi grows

linearly with N. The fluctuations X hXi on the

scale O(N

1=2

) are governed by the central limit

theorem that states that (X hXi)=N

1=2

becomes for

large N a Gaussian random variable with variance

ihyi

  as in [7]. Finally, its fluctuations on

the larger scale O(N) are governed by the large

deviation form [8]. The possible non-Gaussianity of

the y’s leads to a nonquadratic behavior of H

for (large) deviations from the mean, starting from

X hXi=N ’ =S

000

(0). Note that if y is Gaussian,

then X is Gaussian too for any t, but the universal

formula [8] with H = (X  Nhyi)

=2N is valid

only for t  .

Single-Particle Diffusion

For the pure advection without noise, the dis-

placement of the single Lagrangian trajectory is

R(t)  R(0) =

V(s)ds, with V(t ) = v(R(t), t) being

the Lagrangian velocity. One can show that V(t)is

statistically stationary in the frame of reference with

no mean flow and under statistical hom ogeneity and

stationarity of the incompressible Eulerian velocities.

For  = 0, the mean square displacement satisfies the

equation

h½RðtÞRð0Þ

i¼2

hVð0ÞVðsÞids ½9

The behavior of the displacement is crucially

dependent on the Lagrangian correlation time  of

V(t) defined by

hVð0ÞVðsÞids ¼hv

i ½10

No general relation between the Eulerian and

the Lagrangian correlation times has been estab-

lished, except for the case of short-correlated

velocities. For times t  , the two-point function

in [9] is approximately equal to hV(0)

i= hv

The fluid particle transport is then ballistic with

h[R(t)  R(0)]

i’hv

and the PDF P(R, t)is

determined by the whole single-time velocity PDF.

When the correlation time of V(t) is finite (a generic

situation in a turbulent flow where  is of order of a

large-scale turnover time), an effective diffusive regime

is expected to arise for t   with h(R(t)  R(0))

i’

2hv

it. Indeed, the particle displacements over time

segments much larger than  are almost independent.

At long times, the displacement R(t) behaves then as a

sum of many independent variables and falls into the

class of stationary processes treated in the previous

section. In other words, R(t)fort   becomes

a Brownian motion in d dimensions, normally

distributed with hR

(t)R

(t)i’D

t, where the

so-called eddy diffusivity tensor is as follows:

ð0ÞV

ðsÞþV

ð0ÞV

ðsÞids ½11

The symm etric second-order tensor D

is the only

characteristics of the velocity which matter s in this

limit of t  . The trac e of the tensor is equal to

i, that is, equal to the large-time value of the

integral in [9], while its tensor ial pro perties reflect

the rotational symmetries of the advecting velocity

field. If the latter is isotropic, the tensor reduces to a

diagonal form characterized by a single scalar value

. The main problem of turbulent diffusion is to

obtain the effective diffusivity tensor given the

velocity field v and the value of the molecular

diffusivity .

Two-Particle Dispersion in Smooth

Flows

Even when velocity v(R, t) is a smooth function of

the coordinates, Lagrangian dynamics can be quite

256 Lagrangian Dispersion (Passive Scalar)