Eschrig H. Topology and Geometry for Physics

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Unitarity means 1

¼ g

g ¼ð

l¼0

ðu

Þ1

þ i

ijk

123

ijk

; where the last

sum again vanishes, leaving

ðu

¼ 1 as the unitarity condition. The con-

straint det g ¼ 1 yields again the same condition. (It was already plugged into

(8.42) by the traceless Pauli matrices, recall det ðexp AÞ¼expðtr AÞ:) Hence,

SUð2Þ is homotopic to the sphere S

: Put r

¼ i1

, then the inverse relations are

ðgÞ¼tr ðgr

Þ=ð2iÞ; ð8:44Þ

that is, the mapping SUð2Þ!S

is even a bijection. (Distinct from (6.53, 6.54),

here the sphere S

in the Euclidean space R

ﬁgures.) The parameter space S

3 u

is of course to be distinguished from the base space R

3 x

of the principal ﬁber

bundle, in which the gauge ﬁelds live.

Returning to the base space R

; take the two patches U

¼fðx

Þ2

jjxj\2Rg and U

¼fðx

Þ2R

jjxj[ R=2g of R

for some ﬁxed radius R,

and gauge away the pure gauge outside R=2, that is, ﬁx the local gauge potential

¼ 0 )A

¼ w

1

ðow=ox

Þ; ð8:45Þ

where w ¼ w

ðgðx

ÞÞ is the transition function from U

to U

: It sufﬁces to

consider the transition function for jxj¼R, that is, on another sphere S

: In order

to classify possible ﬁeld conﬁgurations one has to classify the mappings S

ðx

Þ7!gðx

Þ2SUð2ÞS

: This is provided by the homotopy group p

ðSUð2ÞÞ ¼

ðS

Þ¼Z:

Use homogeneous coordinates ðw

Þ¼ðx

Þ=R; j wj¼1, on the sphere S

of the

base space and consider the mappings S

! S

SUð2Þ : ðw

Þ7!ðu

Þ¼

ðtr ðg

ðw

Þr

Þ=ð2iÞÞ ¼ ðtr ððw

þ w

Þr

Þ=ð2iÞÞ; n 2 Z

: For n ¼ 1 this is

the identity mapping as seen from (8.44). For n [ 1, the sphere S

is ‘wrapped n

times’ by its preimage S

as can be seen from the last relation (8.43) since g 7!g

corresponds to t

7!nt

, and the mapping g 7!ðu

Þ is bijective. As is likewise easily

seen for small t

, the mappings preserve orientation of the manifold S

in the

vicinity of its north pole ðu

¼ 1Þ, and hence everywhere due to smoothness. Let

j : S

! S

be the mapping which interchanges the coordinates w

and w

and

hence reverses orientation of S

: Replacing above g

ðw

Þ by g

ðjðw

ÞÞ yields

representatives for negative integer homotopy classes. All mappings for non-zero n

are not homotopic to the trivial mapping g

ðw

Þ¼e ¼ 1

Belavin and Polyakov considered (anti)self-dual solutions F¼Fof the

Yang–Mills equations. Under this condition the ﬁeld part of the Yang–Mills action

becomes ð2k

1

tr ðF ^ FÞ; where the integrand is a 4-form in a four-

dimensional space and hence is closed, dtr ðF ^ FÞ ¼ 0: Since the patch U

contractible, tr ðF ^ FÞ is also exact (end of Sect. 5.5), that is, there exists a

3-form K so that tr ðF ^ FÞ ¼ dK: (Recall that F¼0 on U

and hence this

relation is trivially fulﬁlled on U

with any constant K:)

8.4 Gauge Fields and Connections on Manifolds 267

In the given case,

K ¼ tr A^dAþ

A^A^A



; dK ¼ tr ðF ^ FÞ: ð8:46Þ

and

tr ðF ^ FÞ ¼

dK ¼

K ¼

tr ðA^A^AÞ: ð8:47Þ

Proof Straightforwardly, with dA¼FA^A;

dK ¼tr dA^dAþ

ðdA^A^AA^dA^AþA^A^dA



¼tr ðF A^AÞ^ðF A^AÞð

ðF A^AÞ^A^AA^ðF A^AÞ^AþA^A^ðF A^AÞðÞ



¼tr F ^F F ^A^AA^A^F þA^A^A^Að

ðF ^A^AA^F ^AþA^A^F A^A^A^AÞ



Now, using the alternating property of the wedge product and the cyclicity of the

trace of matrices,

tr ðA^A^A^AÞ¼

jklm

tr ðA

Þdx

^ dx

¼

jklm

tr ðA

Þdx

^ dx

¼tr ðA^A^A^AÞ:

Hence, tr ðA^A^A^AÞ¼0: Likewise, tr ðF^A^AÞ¼tr ðA^F^AÞ¼

tr ðA^A^FÞis found. This reduces the above result for dK to (8.46) and thus

proves the latter. In the next section this will become a special case of a very

general result.

In (8.47), S

is the sphere of radius R in R

; and Stokes’ theorem was used in the

second equality. Again using dA¼FA^A; K may be cast into K ¼ tr ðA ^

FÞ ð1=3Þtr ðA^A^AÞ: The ﬁrst term vanishes on S

, since F¼0 there. h

Now, consider the strength of the instanton,

q ¼

tr ðF ^ FÞ: ð8:48Þ

For the sake of simplicity the identical (sometimes called fundamental) two-

dimensional representation of the symmetry group is considered, wðgÞ¼g, where

g is the ð 2  2Þ-matrix (8.42). The results are qualitatively general, only the

268 8 Parallelism, Holonomy, Homotopy and (Co)homology

topological charge may get an additional dimension factor from the trace due to a

more general representation. For g

ðx

Þ¼e, from (8.45) one has A

¼ 0 and

hence q ¼ q

¼ 0: For g

ðx

Þ¼w

; one has A

¼ðw

1

ðoðw

Þ=ox

Þ¼ðw

1

ðoðw

Þ=

Þ=R: Since this corresponds to the identity mapping S

7!u

ðw

Þ¼w

, the gauge potential may be expressed as A

ðu

Þ¼g

1

ðog=ou

Þð1=RÞ¼ð1=RÞðolng=ou

Þ¼ði=RÞ

ðot

=ou

Þ or A

i;l

ðot

=ou

Þdu

¼i

: This yields trðA

Þ¼i

tr ðr

Þdt

^dt

¼i

3!trð r

Þdt

^dt

¼3! 2ds; where ds is the 3-volume ele-

ment of the manifold SUð2Þ: The integration of the last expression of (8.47) is

now replaced by an integration over the unit sphere S

SUð2Þ with the result

q ¼q

¼4 2p

¼8p

where 2p

is the volume of the unit sphere S

(see

footnote on p. 53). Now, realize that g

¼g

n1

: Since the gauge is a pure gauge

on S

, it can be gauged away on every trivial patch (chart) of S

: Cover S

by U

and U

, the north and the south open hemisphere overlapping at the equator.

Gauge away g

n1

from the north hemisphere, that is, deform g

n1

ðw

Þ smoothly

into g

n1

ðw

Þ, where g

n1

¼e is constant on the north hemisphere. In view of the

bijection between SUð2Þ and S

this amounts to a smooth coordinate transfor-

mation on S

, which transforms the integral over S

nðnorth poleÞ into an integral

over the south hemisphere. Likewise gauge away g

from the south hemisphere.

Now, g

¼eg

¼g

on the north hemisphere and g

¼g

n1

e ¼g

n1

on the south

hemisphere, and ð1=3Þ

trðA

Þ¼q

þq

n1

: For negative n, the

reversion of orientation of S

simply results in q

n

¼q

: In summary,

¼8p

n or

n ¼

24p

tr g

1

dg ^ g

1

dg ^ g

1





;

ð8:49Þ

that is, the Belavin–Polyakov instanton has a topologically quantized strength

(topological charge, cf. Sect. 2.6). Note, that compared to Dirac’s monopole there is

not even a singularity string of the gauge potential in the present case; the gauge

potential of the Belavin–Polyakov instanton is smooth everywhere in the base space

: It is a soliton-like solution of the ﬁeld equations, which per se also has no length

scale: R was arbitrary in the choice of U

and U

,(8.48) does not depend on it.

However, distinct from an ordinary soliton the instanton ﬁeld strength F is non-

zero in a vicinity of the origin of the four-space only: it is local and exists only an

instant of time, hence instanton. Its presence imposes a gauge invariant non-trivial

boundary condition for ﬁelds propagating in time from 1 to 1:

Recall that in this case the quantization of (8.49) had its origin in the

requirement that the gauge ﬁeld vanishes at inﬁnity, or the gauge potential is pure

there. In fact, instead of R

the compactiﬁed space R

S

was treated.

8.4 Gauge Fields and Connections on Manifolds 269

8.5 Characteristic Classes

The topological quantizations (8.37) and (8.49) have a common feature which

reﬂects a very general algebraic structure admitting of a deep analysis. In both

cases an integral of an exact r-form (coboundary) over an r-dimensional closed

manifold (cycle) M is taken as a sum of integrals over charts, the items of which

are transformed using Stokes’ theorem into integrals over the boundaries of the

charts. If the continuation of the preimage of the coboundary from one chart into

the other is obstructed, then the total integral may be nonzero, but only depending

on a cohomology class of H

ðMÞ, called a characteristic class. (Compare also

Sect. 8.2.) Recall, that a gauge potential is a local connection form on a principal

ﬁber bundle and the gauge ﬁeld is its local curvature form.

Let ðP; M; p; GÞ be a principal ﬁber bundle with the Lie group G as its structure

group and g as its Lie algebra. In view of possible ﬁber bundles ðE; M; p

; F; GÞ

associated with P; G and g may be replaced in the following by any complex

matrix representation in F: An Ad G invariant symmetric r-linear function p :

g gðr factorsÞ!C is a symmetric ðpð...; X

; ...; X

; ...Þ¼pð...; X

; ...;

; ...ÞÞ r-linear function with the property

pðAd gX

; ...; Ad gX

Þ¼pðX

; ...; X

Þ for all g 2 G; X

2 g: ð8:50Þ

In matrix representations,

pðgX

1

; ...; gX

1

Þ¼pðX

; ...; X

Þ: ð8:51Þ

Given a connection form x on P, recall that the curvature form X ¼ Dx is a g-

valued tensorial 2-form on P, that is, at every q 2 P; hX; Z

^ Z

i2g for any pair

of tangent vectors Z

; Z

of T

ðPÞ: Deﬁne

ðZ

; ...; Z

Þ¼

ð2rÞ!

ð1Þ

jPj

pðhX; Z

Pð1Þ

^ Z

Pð2Þ

i; ...; hX; Z

Pð2r1Þ

^ Z

Pð2r

ÞiÞ;

ð8:52Þ

where p is any Ad G invariant r-linear complex function, P means permutations of

the numbers ð1; ...; 2rÞ, and Z

2 T

ðPÞ: Then, there holds the

Chern–Weil theorem (a) There is a unique global 2r-form on M equal to the

local 2rforms p

¼ s



ðp

Þ (p. 224) on trivializing neighborhoods U

of the base

space M of P, which is closed: dp

¼ 0:

(b) Let x and x

be two different connection forms on P: Then, the difference

 p

is an exact 2r-form on M, that is, the de Rham cohomology class

associated with the glued together p

in H

ðMÞ is independent of x:

Proof (a) As deﬁned in Sect. 7.5, the local curvature form X

is uniquely deﬁned

by X and is linearly depending on X for every coordinate neighborhood U

M : hX

i¼hs



ðX

ðxÞ

Þ;X

i¼hX

ðxÞ

a

ðX

Þ^s

a

ðX

Þi¼hX

ðxÞ

;

270 8 Parallelism, Holonomy, Homotopy and (Co)homology

ðs

a

ðX

ÞÞ^

ðs

a

ðX

ÞÞi: By r-linearity, p

¼s



ðp

Þ is uniquely deﬁned by p

through Z

¼s

a

ðX

Þ for which inversely p



ðZ

Þ¼X

; where p is the bundle

projection of the principal ﬁber bundle P: The tangent vector Z

ðxÞ

may be pushed

forward to any other point s

ðxÞg on the ﬁber p

1

ðxÞ as ðR



ðxÞ

; and

hX;ðR



^ðR



i¼hR



X;Z

i¼Adðg

1

ÞhX;Z

i; since X is a ten-

sorial form of type ðAd;gÞ: Since ðR



ðxÞ

Z

ðxÞ

is vertical, also



ððR



ðxÞ

Þ¼X

; and in summary hp



ðZ

Þ;...;p



ðZ

Þi¼hp



ðp

Þ;Z

;...;

i¼hp

;...; Z

i or in short p



ðp

Þ¼p

; where X

¼p



ðZ

Þ¼p



Þ (since



ZÞ¼0). This also implies that p

¼p

on U

, and hence the local

forms p

deﬁne a unique global 2r-form on all M, which is pulled back to p

on P

by the bundle projection.

Next it is shown that for every n-form p on P which is equal to p



p for some

n-form

p on M it holds that dp ¼ Dp: Indeed, again with X

¼ p



ðZ

Þ¼p



and by linearity of p



and d; ðdpÞðZ

; ...; Z

nþ1

Þ¼ðdp



pÞðZ

; ...; Z

nþ1

Þ¼ðp



pÞ

ðZ

;...;Z

nþ1

Þ¼ðd

pÞðX

;...;X

nþ1

Þ¼ðp



pÞð

;...;

nþ1

Þ¼ðdpÞð

;...;

nþ1

Þ¼

ðDpÞðZ

;...;Z

nþ1

Þ:

Now, from Bianchi’s identity, 0 ¼

pð...; hDX; Z

^ Z

i; ...Þ¼Dpð...;

hX; Z

^ Z

i; ...Þ¼Dp

ðZ

; ...; Z

2rþ 1

Þ¼dp

ðZ

; ...; Z

2rþ 1

Þ¼dp

ðX

; ...; X

2rþ1

Þ;

the last equality again by linearity of p



and d: This completes the proof of (a).

(b) Let x

and x

be two connection forms on P, that is, two g-valued 1-forms

with properties 1 and 2 given on p. 215. In view of the afﬁne linearity of 1 and the

linearity of 2 of these properties in x; x

¼ x

þ th; h ¼ x

 x

, is another

connection form for every t 2½0; 1, and X

¼ dx

þ½x

; x

¼dx

þ½x

; x

þ

tðdh þ½x

; hþ½h; x

Þ þ t

½h; h¼X

þ tðdh þ½x

; hþ½h; x

Þ  t

½h; h¼X

h  t

½h; h is the corresponding curvature form. One has dX

=dt ¼ D

Moreover,

 p

dtdp

=dt ¼ r

dtpðhD

h; ^i; hX

; ^i; ...; hX

;  ^ iÞ

¼ r

dtD

pðhh; i; hX

; ^i; ...; hX

;  ^ iÞ

¼ r

dtdpðhh; i; hX

; ^i; ...; hX

;  ^ iÞ

¼ d

dtrpðhh; i; hX

; ^i; ...; hX

;  ^ iÞ ¼ dH:

ð8:53Þ

8.5 Characteristic Classes 271

The ﬁrst equality is trivial, in the second the symmetry of p

was used, in the third

Bianchi’s identity D

¼ 0 was exploited, and in the fourth it was realized that D

again applies to a pull back from M by p since h like the connection forms x

is a

pseudo-tensorial form of type ð Ad; gÞ; pulled back from its local form on M by p:

Now, since H, the integral of the last line, is a pseudo-tensorial form, in analogy to

(a) a form H

¼ s



ðHÞ may be deﬁned, so that p

 p

¼ dH

on M:

According to (5.39), the de Rham cohomology classes, that is, the group elements

of the de Rham cohomology group H

ðMÞ are the sets of closed 2r-forms which

differ at most by an exact 2r-form. Hence, p

and p

belong to the same

element of H

ðMÞ. h

As in Sect. 8.2, the de Rham cohomology classes associated with p

are called

the characteristic classes. They depend on P and on the chosen Ad G invariant

r-linear function p, but not on the chosen connection on P:

The set of formal sums of Ad G invariant symmetric r-linear functions (for all

integer r 0, complex numbers for r ¼ 0) is made into a graded commutative

algebra I



ðGÞ by deﬁning the product

ðX

; ...; X

rþs

Þ¼

ðr þ sÞ!

pðX

Pð1Þ

; ...; X

PðrÞ

Þp

ðX

Pðrþ1Þ

; ...; X

PðrþsÞ

Þ:

ð8:54Þ

Note that r-linear functions by (8.52) give rise to forms of even degree 2r:

Weil homomorphism The mapping I



ðGÞ!H



ðMÞ by p 7!fp

g is a

homomorphism of graded algebras.

g means the de Rham cohomology class of p

: This result is clear from the

above, and by realizing that the image of the homomorphism consists of coho-

mology groups of even degree only and that in H



ðMÞ the ^-product of factors of

even degree is commutative. Of course, the homomorphism depends on the topology

of M: Hence, the whole mapping depends on the principal ﬁber bundle ðP; M; p; GÞ:

There is a one–one correspondence between symmetric r-linear functions and

polynomials of degree r: Deﬁne the polynomial p

ðrÞ

associated with p by

ðrÞ

ðuÞ¼pðu; ...; uÞ; r arguments; ð8:55Þ

then pðu

; ...; u

Þ is ð1=r!Þ times the coefﬁcient of t

t

in p

ðrÞ

ðt

þþt

Þ;

this is called the polarization of the polynomial p

ðrÞ

: It is clear that, iff

pðX

; ...; X

Þ is Ad G invariant, so is p

ðrÞ

, it is called an Ad G invariant polyno-

mial. Now, I



ðGÞ is isomorphic with the algebra of Ad G invariant polynomials.

An in a sense most general case is a complex vector bundle

ðE; M; p

; C

; Glðk; CÞÞ on a (real) m-dimensional base manifold M, associated

with the principal ﬁber bundle ðP; M; p; Glð k; CÞÞ: In this case there are k distinct

Ad G invariant polynomials obtained from the characteristic polynomial of a

general complex ðk kÞ-matrix X as an element of glðk; CÞ:

272 8 Parallelism, Holonomy, Homotopy and (Co)homology

det k1



2pi



r¼0

ðrÞ

ðXÞk

kr

: ð8:56Þ

The (in principle arbitrary) normalization convention of X ensures that the Chern

numbers deﬁned below will be real integers or fractions with small integer

denominators. Since det ðk

 gXg

1

Þ¼det ðgðk

 XÞg

1

Þ¼det g detðk



XÞdet ðg

1

Þ¼det ðk

 XÞdet ðgg

1

Þ¼det ðk

 XÞ; it is clear, that the

polynomials p

ðrÞ

ðXÞ are Ad G invariant. Let X be the curvature form of some

connection form x on P: The rth Chern class C

ðEÞ of the complex vector bundle

E is the de Rham cohomology class of the closed 2r-form

ðY

; ...; Y

Þ¼p

ðs

a

ðY

Þ; ...; s

a

ðY

ÞÞ; Y

2XðMÞ; ð8:57Þ

where p

ðZ

; ...; Z

Þ is the polarization of p

ðrÞ

ðZÞ: After introducing a base in

ðMÞ; x 2 U

, the 2-form X

becomes a complex ðk  kÞ-matrix. A somewhat

involved but straightforward calculation yields

ð1Þ

ð2piÞ

ðr!Þ

...i

...j

ðX

^^ðX

: ð8:58Þ

Each matrix element ðX

is a 2-form on M: It can be shown that the Chern

classes generate the whole image of I



ðGlðk; CÞÞ in H



ðMÞ: Their representation

depends on E as seen from (8.58). The total Chern class corresponds to the direct

sum over r of the c

Some important other characteristic classes are:

Chern character Consider instead of (8.56) the expression

tr exp 

2pi



¼ tr

r¼0



2pi



r¼0

ðrÞ

ðXÞ: ð8:59Þ

It is easily seen that because of the trace the left expression is Ad G invariant.

The rth Chern character Ch

ðEÞ corresponds to p

ðrÞ

, in a base of T

ðMÞ,

¼ð1=r!Þtr ðiX

=ð2pÞ^^iX

=ð2pÞÞ: ð8:60Þ

(ch

is related to p

chX

, the polarization of p

ðrÞ

, like (8.57).) The total Chern

character is again the direct sum, which is ﬁnite, since the last expression vanishes,

if 2r [ dim M:

Todd classes Let E be the Whitney sum E ¼ L

L

, where each L

the complex line bundle over M: Let x

¼ c

ðL

Þ be the ﬁrst Chern class of L

: The

2r-form of the expansion of

td ¼

i¼1

ð^Þ

1  expðx

; ð8:61Þ

8.5 Characteristic Classes 273

where the exponential is meant as the formal ^-expansion, is the rth Todd class

ðEÞ:

Pontrjagin classes They are the analogue of Chern classes for real vector

bundles ðE; M; p

; R

; Glðk; RÞÞ: Instead of (8.56) one uses

det k1





r¼0

ðrÞ

ðXÞk

kr

: ð8:62Þ

Replacing X in the determinant by the skew-symmetric matrix X ¼X

, one ﬁnds

det ð1

 XÞ¼det ð1

þ X

Þ¼det ð1

þ XÞ since det A ¼ det A

: From that it

immediately follows that the odd Pontrjagin classes vanish. One ﬁnds

ð2pÞ

ð2rÞ!

...i

...j

ðX

^^ðX

: ð8:63Þ

If one identiﬁes the complex k-vector bundle E with the real 2k-vector bundle E

then the rth Chern class becomes the 2rth Pontrjagin class, C

ðEÞ¼P

ðE

Þ:

Euler class Consider an orientable manifold M of even dimension dim M ¼

2m and let ðTðMÞ; M; p

; R

; Oð2mÞÞ be the tangent vector bundle on M asso-

ciated with the reduced frame bundle ðL

ðMÞ; M; p; Oð2mÞÞ of orthonormal

frames. The Euler class is given by

e ¼

ð1Þ

ð4pÞ

1...2m

...i

ðX

^^ðX

2m1

; ð8:64Þ

in view of X

¼X

implying e

¼ð2pÞ

2m

det ðX

Þ and hence being Ad Oð2mÞ

invariant.

Let ½z be the homology class of a 2r-dimensional cycle in M, and let ½ p be the

cohomology class of a closed 2r-form on M (cocycle). Equation 5.40 says that the

integral

p depends only on the (co)homology classes of z and p, and hence is a

topological invariant. Since characteristic classes are closed forms on M, they give

rise to topological invariant numbers by integration over cycles in M: Best known

is the

Gauss–Bonnet–Chern–Avez theorem Let M be an orientable 2m-dimen-

sional compact manifold, let e be its Euler class and let vðMÞ be its Euler char-

acteristic (Euler–Poincaré characteristic) (5.63). Then,

vðMÞ¼

e: ð8:65Þ

Particular cases of this general theorem are considered in [6]. For 2m ¼ 2 the

theorem reduces to the well known Gauss–Bonnet theorem vðMÞ¼1=ð2pÞ

where K ¼ X

is the curvature form of the surface M:

274 8 Parallelism, Holonomy, Homotopy and (Co)homology

Integrals over a 2r-cycle in M of an rth characteristic class are called Chern

numbers. In ﬁeld theory, for dim M ¼ 4, the Chern numbers

and

^ ch

ð8:66Þ

are of particular interest. In the real case with dim M ¼ 4;

is the Pontrjagin

number.

Let ðE; M; p

; K

; GÞ be a K-vector bundle associated with a principal ﬁber

bundle ðP; M; p; GÞ, and let p

be a 2r-form (8.52) representing a characteristic class

of P in a K-matrix representation. According to the Chern–Weil theorem, p



ðp

Þ for all trivializing neighborhoods U

2 M deﬁnes a closed 2r-form on all M:

In view of Poincaré’s lemma (end of Sect. 5.5) this implies that locally, but not in

general globally, p

is also exact, that is, there are local ð2r  1Þ-forms q

, so that

¼ dq

: ð8:67Þ

Let x ¼ x

be a connection form leading to the curvature form X: On a trivial-

izing subbundle U

G of P, the vertical ‘unit’ form x

which is pulled back to

0 ¼ x

¼ s



ðx

Þ provides a ﬂat connection on U

: Let x

¼ tx

¼ ts



ðxÞ on

: Then, after pulling back with s



the chain of equations (8.53) in the proof of

the second part of the Chern–Weil theorem yields

¼ r

dtpðhx

; i; hX

; ^i; ...; hX

;  ^ iÞ; ð8:68Þ

where X

¼ tdx

þ t

½x

; x

¼tX

þðt

 tÞ½x

; x

: The g-valued (in the

representation vector space K

) local ð2r  1Þ-form q

on U

2 M is called the

Chern–Simons form of p

Consider as an example the Chern–Simons ð2r  1Þ-form of the rth Chern

character Ch

ðrÞ

cha

ðr  1Þ!



dttr ðx

^ X

^^X

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

r1factors

Þ: ð8:69Þ

In particular

ð1Þ

cha

dttr x

tr x

;

ð2Þ

cha



dttr ðx

^ðtdx

þ t

^ x



tr x

^ dx

^ x



;

 ð8:70Þ

8.5 Characteristic Classes 275

In local gauge ﬁeld theory x

is commonly denoted A as the gauge potential and

is denoted F as the gauge ﬁeld. In a one-dimensional vector bundle (line

bundle) on M of two dimensions one has simply tr A¼A: The vector potential

A ¼ði=eÞA is just ð2p=eÞq

ð1Þ

: Hence, (8.39) is a trivial case of a Chern character

eF=ð2pÞ and a Chern–Simons form eA=ð2pÞ: Likewise it is seen that

ð1=ð8p

ÞÞtr ðF ^ FÞ is a Chern character Ch

, and that K of (8.46)isuptoa

factor of convention a Chern–Simons form: K ¼8p

ð2Þ

: Both relations (8.39,

8.46) are special cases of (8.67).

8.6 Geometric Phases in Quantum Physics

8.6.1 Berry–Simon Connection

Consider a quantum system under the inﬂuence of its surroundings. For the sake of

simplicity non-relativistic quantum mechanics is considered, although more gen-

eral cases could be treated similarly. The system is described by a Hamiltonian,

and the inﬂuence of the surroundings is expressed by a set of in general time-

dependent parameters the Hamiltonian depends on. Collect the parameters into a

set R of real numbers which varies in some real m-dimensional manifold. Let the

Hamiltonian and hence its eigenvalues, calculated for ﬁxed R,

HðRÞjWðRÞi ¼ jWðRÞiEð RÞ; hWð RÞjWðRÞi ¼ 1; R fixed, ð8:71Þ

continuously depend on R [7]. Let EðRÞ be a non-degenerate and isolated eigen-

value of

HðRÞ for some value R of the parameters. Then, a manifold M can always

be found on which EðRÞ varies continuously with R and remains isolated. Since

jWðRÞi is deﬁned up to a phase e

, a Lie group Uð1Þ is attached to each point R of

the manifold M, which makes it into a principal ﬁber bundle ðP; M; p; Uð1ÞÞ : (The

Lie group Uð1Þ is the symmetry group related to the conservation of hWjWi for

complex jWi which eventually is related to particle conservation).

Let R depend on time t through s ¼ t=T where T is a speed scaling factor for

this dependence. The time-dependent Schrödinger equation reads ðh ¼ 1Þ

djW

ðRðsÞ; tÞi

HðRðsÞÞjW

ðRðsÞ; tÞi: ð8:72Þ

Let jWðRðsÞÞi be the state of (8.71). Then, the quantum adiabatic theorem says

that

lim

T!1

ðRðsÞ; tÞihW

ðRðsÞ; tÞj ¼ jWðRðsÞÞihWðRðsÞÞj; ð8:73Þ

where s is kept constant in the limiting process. In order to determine the phase

change of jWðRÞi on a path through M, put the ansatz

276 8 Parallelism, Holonomy, Homotopy and (Co)homology