Marathe K. Topics in Physical Mathematics

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148 5 Characteristic Classes

det



λI

−

2π



=det



λI

2π



Therefore p

= p

= ...=0and p

again corresponds to the kth Pontryagin

class. In Example 5.3, p

2k+1

=0in general but w(p

2k+1

)=0, since every

GL(n, R)-connection is reducible to an O(n, R)-connection.

Example 5.5 With the notation of Example 5.4, we construct an SO(2m)-

invariant polynomial, called the Pfaﬃan, which is not invariant under the

action of O(2m).ThePfaﬃan Pf is the homogeneous polynomial of degree

m such that, for A ∈ so(2m),

Pf(A)=



sgn(σ)A

σ(1)σ(2)

···A

σ(2m−1)σ(2m)

, (5.15)

where the sum is over all permutations of (1, 2,...,2m). The class w(Pf) is

called the Euler class of P (M,SO(2m)) and is the class [γ] such that

∗

(γ)=



sgn(σ)Ω

σ(1)

σ(2)

∧···∧Ω

σ(2m−1)

σ(2m)

. (5.16)

The Euler class occurs as the generalized curvature in the Chern–Gauss–

Bonnet theorem, which states that



Pf(

Ω)=χ(M ). (5.17)

This theorem generalizes the classical Gauss–Bonnet theorem for compact,

connected surfaces in R

.ForsuchasurfacethePf(

Ω) is a multiple of the

volume form on M. The multiplier κ is the Gaussian curvature and the above

theorem reduces to



κ = χ(M).

The expression for the Euler class of a 4-manifold as an integral of a poly-

nomial in the Riemann curvature was obtained by Lanczos [242] in his study

of Lagrangians for generalized gravitational ﬁeld equations. He observed that

the integral was invariant and so did not contain any dynamics, but he did

not recognize its topological signiﬁcance. The general formula for the Euler

class of an arbitrary oriented manifold was obtained by Chern and it was in

studying this generalization that Chern was led to his famous characteristic

classes, now called the Chern classes.

Example 5.6 Let G = GL(n, C). I

(G) is generated by the characteristic

classes c

,...,c

deﬁned by

det



λI

−

2πi





i=0

n−i

5.3 Characteristic Classes 149

The classes w(c

) ∈ H

(M; C), k =1, 2,...,n, are called the Chern classes

of P .

Example 5.7 G = U(n). I

(G) is generated by the classes c

deﬁned in Ex-

ample 5.6. However, for A ∈ u(n) one has the following complex conjugation

condition:

det



λI

−

2πi



det



λI

−

2πi



Therefore the class w(c

) turns out to be a real cohomology class, i.e.,

w(c

) ∈ H

(M; R) ⊂ H

(M; C),whereH

(M; R) is regarded as a sub-

set of H

(M; C) by the isomorphism induced by the inclusion of R into

C. In view of the fact that the GL(n, C)-connection is reducible to a U(n)-

connection we ﬁnd that the classes w(c

) of Example 5.6 are in fact real.

As we have indicated above, the characteristic classes turn out to be real

cohomology classes. Indeed, the normalizing factors that we have used in

deﬁning them make them integral cohomology classes.

The total Chern class c(P )ofP is deﬁned by

c(P ):=1+c

(P )+···+ c

(P ), (5.18)

and the Chern polynomial c(t) is deﬁned by

c(t):=



i=0

n−i

(P )=t

+ t

n−1

(P )+···+ tc

n−1

(P )+c

(P ). (5.19)

We factorize the Chern polynomial formally as follows:

c(t)=(t + x

)(t + x

) ···(t + x

). (5.20)

The Chern classes are then expressed in terms of the formal generators

,...,x

by elementary symmetric polynomials. For example,

(P )=x

+ x

+ ···+ x

(P )=



1≤i<j≤n

(P )=x

···x

It is well known that every symmetric polynomial in x

,...,x

can be ex-

pressed as a polynomial in elementary symmetric polynomials. In particular,

the homogeneous polynomial



i=1

of degree k canbeexpressedinterms

of the ﬁrst k Chern classes. These expressions are useful in many character-

istic classes.

The Chern character ch(P ) is the polynomial in x

deﬁned by

150 5 Characteristic Classes

ch(P )=



i=1

∈ H

∗

(M; Q), (5.21)

where the exponential function on the right hand side is interpreted as a for-

mal power series, which in fact is ﬁnite and terminates after

(dim M)terms.

It is customary to say that the Chern character is deﬁned by the generating

function e

. The ﬁrst few terms on the right hand side, expressed in terms

of the Chern classes, give us the following formula

ch(P )=n + c

(P )+



(P ) − c

(P )



+ ···.

We now give examples of some other important characteristic classes that

can be deﬁned by generating functions and then expressed as polynomials

in x

• The To dd cl ass τ(P ) ∈ H

∗

(M; Q) deﬁned by the generating function

z/(1 − e

−z

)isgivenby

τ(P )=



i=1

1 −exp(−x

)

. (5.22)

Its expression in terms of the Chern classes is given by

τ(P )=1+

+ c

+ ···. (5.23)

The Todd class appears in the statement of a version of the Riemann–Roch

theorem.

• The Hirzebruch L-polynomial deﬁned by the generating function

z/tanh(z)isgivenby

L(P )=



i=1

tanh(x

)

. (5.24)

Its expression in terms of the Pontryagin classes is given by

L(P )=1+

(7p

− p

)+···. (5.25)

The Hirzebruch L-polynomial appears in the Hirzebruch signature theo-

rem, which is discussed later in this chapter.

• The

A genus deﬁned by the generating function z/(2 sinh(z/2)) is given

A(P )=



i=1

2sinh(x

/2)

. (5.26)

5.3 Characteristic Classes 151

Its expression in terms of the Pontryagin classes is given by

A(P )=1−

5760

(7p

− 4p

)+···. (5.27)

The

A genus is a topological invariant of spin manifolds. It is in establishing

the divisibility properties of this invariant that Atiyah and Singer were led

to their famous index theorem.

The characteristic classes satisfy the following properties.

(i) Naturality with respect to pull-back of bundles:

If f : N → M and f

∗

P is the pull-back of the principal bundle P (M,G)

to N ,thenwehave

b(f

∗

(P )) = f

∗

(b(P )), (5.28)

where b(P ) is a characteristic class of the bundle P .

(ii) The Whitney sum rule:

b(E

⊕ E

)=b(E

)b(E

), (5.29)

where E

are vector bundles and b is a characteristic class.

If E is a complex vector bundle with ﬁber C

, which is associated to

P (M, GL(n, C)), then the kth Chern class c

(E)ofE is represented by

w(c

) ∈ H

(M; C). If V is a real vector bundle over M and V

its complex-

iﬁcation, then the kth Pontryagin class p

(V )ofV is given by

(V )=(−1)

) ∈ H

(M; R). (5.30)

We now work out in detail the Chern classes for an SU(2)-bundle P over

a compact manifold M. They are used in deﬁning invariants of isospin gauge

ﬁelds and in particular the instanton numbers.

Let

A =





be a matrix in the Lie algebra su(2). The algebra of characteristic classes

is generated by the images of the invariant symmetric polynomials Tr A =

+ a

and det A = a

− a

.Letω be a connection on P with

curvature Ω.Writinga

= −Ω

/2πi we get the polynomials in curvature

representing the Chern classes

(P )=−

2πi

(Ω

+ Ω

) = 0 (by the Lie algebra property),

(P )=−

4π

(Ω

∧ Ω

− Ω

∧ Ω

)

8π

Tr(Ω ∧ Ω).

152 5 Characteristic Classes

The 2-form

Ω induced by the curvature on the base M will be denoted

by F

. We will see later that F

corresponds to a gauge ﬁeld when ω is

identiﬁed with a gauge potential. Evaluating the second Chern class on the

fundamental cycle of M,weget

(P )[M ]=

8π



Tr(F

∧ F

). (5.31)

In view of the integrality of the Chern classes, the number k deﬁned by

k = −c

(P )[M ] (5.32)

is an integer called the topological charge or topological quantum num-

ber or the instanton number of the principal SU(2)-bundle P .Thetopo-

logical charge may also be deﬁned as p

(P )[M ], where p

is the ﬁrst Pontrya-

gin class. If M is orientable and dim M =4,thenwehave

k = p

(P )=−c

(P ). (5.33)

The characteristic classes described above are often referred to as the pri-

mary characteristic classes. These classes do not depend on the choice

of connection used to deﬁne them and are in fact topological invariants of

the principal bundle P (M, G). They can be obtained by pulling back suit-

able universal classes, as can be shown by using the following theorem due

to Narasimhan and Ramanan [296, 297].

Theorem 5.8 Let G beaLiegroupwithπ

(G) < ∞ and k a positive integer.

Then there exists a principal G-bundle E

,G) with connection θ

,which

is k-universal for principal G-bundles with connection; i.e., for any compact

manifold M with dim M<kand principal bundle P (M,G) with connection

ω there exists a map f : M → B

, deﬁned up to homotopy such that P is the

pull-back of E

to M by f . Then we have the following commutative diagram:

where π, π

are the bundle projections, P = f

∗

,and

f is the lift of f.

Furthermore, we have

ω =(

∗

We note that the spaces B

and E

form two inductive sets and their direct

limits deﬁne topological spaces BG and EG, respectively, such that EG is a

principal G-bundle over BG. The spaces BG, EG are called the universal

classifying spaces for principal G-bundles. They are not ﬁnite-dimensional

5.3 Characteristic Classes 153

manifolds. However, in most applications it is suﬃcient to choose a large

enough k so that the classifying spaces can be taken as ﬁnite-dimensional

manifolds.

Example 5.8 The Stiefel manifold V

(n+k, n) of orthonormal n-tuples of

vectors in R

n+k

is a principal O(n)-bundle over the Grassmann manifold

(n + k, n) of n-planes in R

n+k

. This bundle has a natural connection θ

The bundle V

(n+k, n) with connection θ

is k-universal for principal O(n)-

bundles. Similarly, the principal bundle V

(n+k, n)(G

(n+k, n),U(n)) with

connection θ

is k-universal for principal U(n)-bundles.

This example plays a fundamental role in the proof of Theorem 5.8.

5.3.1 Secondary Characteristic Classes

Primary characteristic classes of principal bundles have become important

tools for deﬁning and analyzing topological invariants in physical theories

and in particular in gauge theories. Recently, another set of characteristic

classes, called the secondary characteristic classes, have appeared in the

Lagrangian formulation of quantum ﬁeld theories. The most well-known of

these classes are the Chern–Simons classes. We now discuss a construction

which leads to another proof of the independence of the primary characteristic

classes of the choice of connections and at the same time prepares the way for

deﬁning secondary characteristic classes. Recall that a standard r-simplex is

deﬁned by



,...,t

) ∈ R

r+1

| t

≥ 0,



i=0



. (5.34)

Let ω

, 0 ≤ i ≤ r, be a set of connections on P (M,G). Let ω =



i=0

Then ω is a connection on P with curvature Ω = d

ω.Letf ∈ I

(G); deﬁne

(ω

,...,ω

):=(−1)

[

r+1

]



f(Ω,...,Ω), (5.35)

where [

r+1

] is the integer part of

r+1

and the integration is along the ﬁber

of the bundle P × Δ

→ P with respect to the standard volume form

∧ ...∧ dt

of Δ

.Thus,Δ

(ω

,...,ω

)isa(2k − r)-form on P ,which

descends to M in view of the fact that f is an invariant polynomial. We

continue to denote this form on M by the same notation. Thus we have a

map

Δ(ω

,...,ω

):I

(G) → Λ

2k−r

(M) deﬁned by f → Δ

(ω

,...,ω

(5.36)

154 5 Characteristic Classes

This map extends to an algebra homomorphism

Δ(ω

,...,ω

):I(G) → Λ(M ), (5.37)

which is called the Bott homomorphism relative to (ω

,...,ω

)[50]. The

following theorem gives an important relation among the forms Δ

Theorem 5.9 Let ω

, 0 ≤ i ≤ r, be a set of connections on P (M, G) and

f ∈ I

(G);then

d(Δ

(ω

,...,ω

)) =



i=0

(−1)

(ω

,..., ˆω

,...,ω

), (5.38)

where ˆ over a variable denotes that this variable is deleted.

Applying the above theorem to the case of a 0-simplex and a 1-simplex we

obtain the following corollaries.

Corollary 5.10 In the case r =0we have

d(Δ

(ω

)) = 0.

Hence, Δ

(ω

) is a closed form and deﬁnes an element of the cohomology

space H

(M; R). The Bott homomorphism induces the homomorphism

[Δ(ω

)] : I

(G) → H

(M; R), deﬁned by f → [Δ

(ω

)]. (5.39)

Corollary 5.11 In the case r =1we have

d(Δ

(ω

,ω

)) = Δ

(ω

) −Δ

(ω

) ∈ Λ

(M,R).

Thus,

[Δ

(ω

)] = [Δ

(ω

)] ∈ H

(M; R),

and hence the homomorphism [Δ(ω

)] of Corollary 5.10 is independent of

the connection ω

and is, in fact, the Chern–Weil homomorphism w deﬁned

earlier.

Corollary 5.11 allows us to deﬁne secondary characteristic classes in the

following way. Let

(G)

(ω)

:= {f ∈ I

(G) | Δ

(ω)=0},

and let

(G)

(ω

,ω

)

:= I

(G)

(ω

)

∩I

(G)

(ω

)

Let f ∈ I

(G)

(ω

,ω

)

; then by Corollary 5.11, d(Δ

(ω

,ω

)) = 0. Hence,

(ω

,ω

) deﬁnes a cohomology class in H

2k−1

(M; R). We call the class

[Δ

(ω

,ω

)] ∈ H

2k−1

(M; R)asimple secondary characteristic class of

thetriple(P, ω

,ω

). We note that in geometrical mechanics and geometric

5.3 Characteristic Classes 155

quantization theory, an important role is played by the Maslov class,which

can be interpreted as a secondary characteristic class. A general discussion of

symplectic geometry and its relation to the secondary characteristic classes

can be found in Vaisman [390].

We now proceed to deﬁne the Chern–Simons classes of P (M,G)with

connection ω (see [75, 76]). The basic tool is the transgression form

(ω) ∈ H

2k−1

(P ; R) deﬁned by

(ω):=−



f(Ω

,...,Ω

)=k



f(ω, Ω

,...,Ω

)dt, f ∈ I

(G),

where



denotes integration along the ﬁber and

:= dt ∧ ω + tΩ + t(1 + t)dω ∈ Λ

(P, g), 0 ≤ t ≤ 1.

Roughly speaking Ω

can be thought of as the “curvature ” form correspond-

ing to the “connection form” tω on P × Δ

. We note that, in general, the

transgression forms do not descend to the base M. However, we have the

following theorem.

Theorem 5.12 Let ω be a connection form on P(M, G).Then

d(T

(ω)) = π

∗

(Δ

(ω)), ∀f ∈ I

(G).

From this theorem it follows that, if f ∈ I

(G)

(ω)

then T

(ω) deﬁnes a

cohomology class [T

(ω)] ∈ H

2k−1

(P ; R), which is called the Chern–Simons

class of (P, ω) related to f ∈ I

(G)

(ω)

. The relation between the secondary

characteristic classes and the Chern–Simons classes is given by the following

proposition.

Proposition 5.13 Let ω

,ω

be two connection forms on the principal bun-

dle P (M,G) and let f ∈ I

(G)

(ω

,ω

)

. Then

∗

[Δ

(ω

,ω

)] = [T

(ω

)] − [T

(ω

)].

An interesting application of the transgression forms is the following de-

scription of the de Rham cohomology ring of an important class of principle

bundles.

Theorem 5.14 (Chevalley) Let G be a compact, connected, semi-simple Lie

group of rank r (dimension of maximal torus in G)andM acompactman-

ifold. Let ω be a connection form on the principal bundle P (M, G).Then

the ring I(G) of invariant polynomials is generated by a set of r elements

,...,f

and the de Rham cohomology H

∗

(P ; R) is given by the quotient

ring

∗

(P ; R)=A/dA,

where

156 5 Characteristic Classes

A = Λ(M )[T

(ω),T

(ω),...,T

(ω)]

is the ring of polynomials in [T

(ω),T

(ω),...,T

(ω)] with coeﬃcients in

Λ(M), the ring of forms on M (pulled back to P bythebundleprojection).

If (M,g) is a Riemannian manifold, P = L(M) the bundle of frames of M

with structure group GL(m, R)andλ the Levi-Civita connection on P ,then

we have the following result.

Theorem 5.15 Let f ∈ I

(GL(m, R)).Letg



be a Riemannian metric

conformal to g and let λ



the corresponding Levi-Civita connection on P .

Then there exists a form W ∈ Λ

4k−1

(P ) such that

(λ



) −T

(λ)=dW. (5.40)

It can be shown that the form W in equation (5.40) can be expressed in

terms of the Weyl conformal curvature tensor.

We now consider a special case in which we can associate to f ∈ I

(G)

(ω)

aformα

∈ H

2k−1

(M; R/Z). Recall that the short exact sequence

0 −→ Z

−→ R

−→ R/Z −→ 0

where ι is the inclusion and p is the canonical projection, induces the long

exact sequence in the cohomology of M

···−→H

(M; Z)

−→ H

(M; R)

−→ H

(M; R/Z) −→···

and a similar sequence in the cohomology of P . Using these sequences we

obtain the following theorem.

Theorem 5.16 Let f ∈ I

(G)

(ω)

be such that w(f) is an integral cohomology

class, i.e., w(f) ∈ ι(H

(M; Z)) ⊂ H

(M; R). Then there exists a form α

∈

2k−1

(M; R/Z) such that

p[T

(ω)] = π

∗

(α

) ∈ H

2k−1

(P ; R/Z).

It is this form α

∈ H

2k−1

(M; R/Z)thatappearsastheChern–Simons

term in the Lagrangian of ﬁeld theories on manifolds with boundary. If A

denotes the pull-back to M of the connection form ω by a local section of

P ,andF the corresponding curvature form on M (in the physics literature,

A is called the gauge potential and F the gauge ﬁeld on M), then the ﬁrst

three Chern–Simons terms have the following local expressions:

2π

tr A,



2π



tr(A ∧ F +

5.4 K-theory 157



2π



tr(A ∧ F ∧ F +

∧ F +

where

:= A ∧ A ∧ ...∧ A



 

n terms

The secondary characteristic classes and the Chern–Simons classes are also

used in describing anomalies such as chiral and gravitational anomalies in ﬁeld

theories. Using quantum ﬁeld theory on a 3-manifold with the Lagrangian

function given by the Chern–Simons term α

, Witten obtained a physical

interpretation of the Jones polynomial of a link by expressing it as the ex-

pectation value of a quantum observable. This work ushered in topological

quantum ﬁeld theory (TQFT) as a new area of research in physical mathe-

matics. We discuss this work in Chapter 11.

5.4 K -theory

The foundations of K-theory were laid by A. Grothendieck in the framework

of algebraic geometry, in his formulation of the Riemann–Roch theorem, pro-

viding a new and powerful tool that can be regarded as a generalized co-

homology theory. Grothendieck’s ideas have led to other K-theories, notably

algebraic and topological K-theories. Grothendieck (b. 1928) was awarded a

Fields Medal at the ICM 1966, in Moscow, for work that gave a new unifying

perspective in the study of geometry, number theory, topology, and complex

analysis. The Grothendieck Festschrift, celebrating his 60th birthday, con-

tains articles on his incredible achievements, which opened up many new

ﬁelds of research in mathematics. Volume 3 (1989) of the journal K-Theory

contains two articles on Grothendieck’s work in K-theory, one by Serre and

the other by his mentor Dieudonn´e. (For a recent update on topological

and bivariant K-theoriesseethebookbyCuntzetal.[92].) Grothendieck

unveiled his work in the ﬁrst lecture at the ﬁrst Arbeitstagung, organized

by Friedrich Hirzebruch

in Bonn in 1957. At the 2007 Arbeitstagung, cele-

brating 50 years of these extraordinary meetings, Hirzebruch gave the ﬁrst

lecture, oﬀering his point of view on the ﬁrst Arbeitstagungen, with special

emphasis on 1957, 1958, and 1962. He explained that Grothendieck lectured

for twelve hours spread out over four days on “Koh¨arente Garben und ver-

allgemeinerte Riemann–Roch–Hirzebruch-Formel auf algebraischen Mannig-

faltigkeiten”

. In these lectures he proved a far reaching generalization of the

Hirzebruch is the founding director of the Max Planc k Institute for Mathematics in Bonn.

This institute and the Arbeitstagung hav e had a very strong and wide ranging impact on

mathematical research since their inception.

Coherent sheaves and generalized Riemann–Roch–Hirzebruch formula for algebraic man-

ifolds.