Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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112
Chapter
6.
Fibre
Bundles
6.3
Vector
Bundles
and
Principal
Bundles
In
many
of
the
above examples,
the
fibre
F
is a
vector
space. Such fibre
bundles
are
special
and
have a name:
Definition:
A vector bundle is a fibre
bundle
whose fibre F is a
vector
space.
As
one
more
example
related
to
manifolds, consider
the
orthonormal
frames
at
a
point.
The
bundle
obtained
by
taking
the
orthonormal
frames
{e
a
}
at
each
point
of
M
is called
the
orthonormal (tangent) frame bundle.
This
is
not
a vector
bundle! To specify a
point
in
the
fibre,
namely
a frame,
one
has
to
specifiy
an
O(n)
rotation
relative
to
some given frame.
Thus
the
fibre is isomorphic
to
the
group
O(n);
it
is
the
group
manifold
of
O(n).
The
structure
group
ofthis
bundle
is also clearly
O(n),
since across
patches
one
has
to
specify
an
O(n)
rotation
to
match
the
fibres. So
in
this
case,
the
fibre is
the
same
as
the
structure
group.
This
type
of
bundle
is also special
and
has
a name:
Definition:
A principal bundle is a fibre
bundle
whose fibre F is
the
structure
group
G
itself.
The
example
of
a helix over
Sl
that
we
encountered
earlier is
another
principal
bundle,
this
time with
fibre
and
structure
group
isomorphic
to
71.,
the
group
of
integers
under
addition.
The
last
few examples have involved
base
spaces
and
fibres which
are
ac-
tually
differentiable manifolds.
They
are
therefore
known
as differentiable fibre
bundles.
Definition:
A differentiable fibre bundle is a fibre
bundle
(E,
X,
F,
7f,
¢,
G) for
which:
(i)
The
base
space
X is
an
n-dimensional differentiable manifold.
(ii)
The
fibre
F
is
an
m-dimensional
differentiable manifold.
(iii)
The
total
space
E
is
an
(m
+
n)-dim.
differentiable manifold.
(iv)
The
projection
7f:
F
--+
X is a
Coo
map,
of
rank
n
everywhere.
(v)
¢
is a collection
of
diffeomorphisms
(rather
than
just
homeomorphisms).
(vi)
G
is a Lie
group
which
acts
differentiably
and
effectively.
(Thus
the
map
g :
F
--+
F
with
g
EGis
a
Coo
map.)
For
a
Coo
manifold,
the
tangent,
cotangent
and
orthonormal
frame
bundles
are
all differentiable fibre bundles.
For
a
trivial
fibre
bundle
(a
direct
product
of
the
base
space
with
the
fibre),
one could define a function
on
the
base
space
taking
values
in
the
fibre.
This
corresponds
to
choosing a specific
point
in
the
fibre above each
point
of
the
base
space.
For
non-trivial
bundles
we
can
still
do
this
locally,
but
not
necessarily
globally.
Definition:
A local section (or local cross-section) of a fibre
bundle
is a contin-
uous injective
map
a:
U
C
X
--+
E
such
that
7f.
a(x)
=
x.
6.3.
Vector
Bundles
and
Principal
Bundles
113
To define a local section is
to
continuously pick a
point
in
E
"above" every
point
x
in a local region of
the
base
space.
The
fact
that
the
image
is "above"
x
is
imposed
by
the
requirement
that
if
we
send
the
image
back
down
to
X
by
the
projection
map
71",
then
it
lands
on
the
original
point
x.
Thus
a local section is
precisely a function
on
an
open
set
of
the
base
space,
with
values
in
the
fibre.
Definition:
A
global section
is a
continuous
injective
map
a:
X
--+
E
such
that
71"'
a(x)
=
x.
This
differs from a local section in being defined all over
the
base
space,
and
not
just
on
an
open
set
of
it.
Local sections always exist,
but
global sections
may
not.
It
is easy
to
convince oneself
that
on
the
helix
there
is
no
global section,
but
on
the
Mobius
strip
there
is.
The
latter
is
illustrated
in Fig. 6.6. After identifying
the
sides
of
the
rectangle
appropriately
with
the
arrows aligned,
the
wavy line
in
the
figure
is a global section
of
the
Mobius
strip
viewed
as
a
bundle
over 8
1
.
Figure
6.6: A global section
of
the
Mobius strip.
A global section
of
the
tangent
bundle
is a
continuous
Coo
vector
field all
over
the
manifold, while a global section
of
the
cotangent
bundle
is a
Coo
I-form
all over
the
manifold.
It
is
easy
to
see
that
such
sections
always
exist. Indeed,
since
the
structure
group
C
=
CL(n,
lR)
acts
as
a linear homogeneous
trans-
formation,
the
zero
vector
field is invariant,
and
one
can
define
it
everywhere.
Similarily
the
I-form
w
=
0 is a global section
of
the
cotangent
bundle.
Whether
there
exist
continuous
everywhere non-zero
sections
of
the
tangent
bundle
is
another
story.
This
depends
on
the
base
manifold.
For
the
two
sphere
8
2
,
there
is a well-known
theorem
sometimes
stated
as follows: "You
can't
comb
the
hair
on
a billiard ball."
If
a 2-sphere were
represented
as
a
ball
and
vector
fields
as
"hair"
attached
at
each
point
of
the
ball,
then
"combing"
the
hair
would
amount
to
flattening
this
set
of
vectors of nonzero
length
to
point
tangentially
to
the
surface.
This
would
be
an
everywhere nonzero
vector
field.
In
fact
it
is
easy
to
convince oneself
that
this
is impossible.
There
is always some
point
on
the
ball where
the
"hair"
has
no
possible
direction
consistent
with
continuity.
In
other
words, 8
2
admits
no
continuous, everywhere non-zero
vector
field.
Exercise:
Try
to
find a
more
precise
proof
that
there
is
no
continuous nowhere-
vanishing vector field
on
8
2
.
One
would like
to
have some
criteria
to
decide
whether
a given
bundle
is
114
Chapter
6.
Fibre
Bundles
trivial
or
not,
since
it
is really in
the
latter
case
that
they
are
most
interesting.
We will
quote
a few
theorems
without
proof.
Theorem:
Given a fibre
bundle
with
base space
X
and
structure
group
G,
if
either X
or
G is contractible
then
the
bundle
is trivial.
(The
converse is
not not
true! A
bundle
may
be
trivial even
though
X
and
G
are
non-contractible.)
This
theorem
has
a corollary which is useful even when
the
spaces
are
not
completely contractible.
This
says
that
we
can
replace
G
by
any
subgroup
of
the
same
homotopy
type,
and
similarly for
X,
without
changing
the
topological
character
of
the
bundle.
Note
that
a topological space
with
the
discrete topology is
not
contractible.
To
be
contractible, a space should have
the
property
that
the
identity
map
,
which sends each element
to
itself,
and
the
constant
map,
which sends each
element
to
a fixed element,
are
homotopic.
This
is never
true
with
the
discrete
topology.
We have
not
yet arrived
at
any
criterion
that
can
tell us
whether
a given
fibre
bundle
is trivial
or
not.
The
following definition will help
us
find such a
criterion.
Definition:
Given
any
bundle
E
with
fibre
F
and
structure
group
G,
we
can
construct
the
associated principal bundle,
denoted
P(E),
by simply replacing
the
fibre
F
with
the
structure
group
G.
This
is a useful
construction
because
of
the
following result:
Theorem:
The
bundles
P(E)
and
E
are
both
trivial
if
and
only if
P(E)
admits
a global section.
Let us use these
theorems
to
study
some fibre bundles.
Examples:
(i)
The
Mobius
strip.
Here
the
structure
group
G
is
Z2,
whose nontrivial element
acts
as a twisting
on
one
end
of
the
strip
before
it
is glued
to
the
other
end.
The
associated principal bundle therefore
has
a fibre
that
consists
of
precisely
two points. Indeed,
this
bundle
is
just
the
boundary
of the
Mobius strip.
Clearly
the
principal
bundle
has
no global section, so
the
Mobius
strip
is a
non-trivial bundle.
It
should
be
recalled
that
the
Mobius
strip
itself
certainly
does
admit
global sections,
but
that
tells us
nothing
about
its
non-triviality
since
it
is
not
a principal bundle. However, by
the
theorem
above, once we have
shown
that
the
associated principal
bundle
is nontrivial
then
we
can
be
sure
the
same
result also holds for
the
Mobius strip.
(ii)
T(S2),
th
e
tangent
bundle
of
S2.
To show
its
nontriviality, we will use
the
fact, referred
to
above,
that
there
is no everywhere non-zero vector field
on
S2.
The
associated principal
bundle
to
T(M)
is
the
bundle
whose fibres
are
GL(n
,
R).
This
is called
the
frame bundle (note
that
it
is
not
orthonormal!),
and
is
denoted
F(M).
Now
an
element
of
GL(n,
R)
can
be
specified
by
a
6.3. Vector
Bundles
and
Principal
Bundles
115
Figure
6.7:
The
principal
bundle
associated
to
the
Mobius
strip.
collection
of
n
linearly
independent,
non-zero real
n-vectors
(these
just
make
up
the
columns
of
the
matrix).
The
existence
of
a global section
of
the
principal
bundle
associated
to
T(M)
thus
reduces
to
the
existence
of
a collection
of
n
independent
vector
fields which
are
everywhere non-zero.
But
no
such
vector
field exists
on
8
2
,
so
F(8
2
)
and
hence also
T(8
2
)
are
non-trivial
bundles.
One
can
contract
GL(n,
R)
to
O(n),
from which we get
the
additional
result
(using
the
corollary
to
a
theorem
above)
that
the
orthonormal
frame
bundle
is
non-trivial
on
8
2
.
Figure
6.8: A non-zero continuous vector field
on
8
1
.
Definition:
A manifold
M
for which
the
orthonormal
frame
bundle
has
a global
section is
said
to
be
parallelisable.
Thus,
the
tangent
bundle
T(M)
to
a manifold
M
is a
trivial
bundle
if
and
only
if
M
is parallelisable.
Example:
8
1
is parallelisable. In fact we only need
one
non-zero
continuous
tangent
vector field, which is
depicted
in Fig. 6.8. Hence,
T(8
1
)
is trivial,
and
is
in
fact
the
infinite cylinder
8
1
x
R.
We conclude
with
a
result
which is
certainly
not
easy
to
prove,
but
is
quite
remarkable.
Theorem:
The
only
compact
manifolds which
are
parallelizable
are
the
man-
ifolds
of
all
the
compact
Lie
groups
and
in
addition
the
7-sphere
8
7
.
(Note
116
Chapter
6.
Fibre Bundles
that
among
the
spheres
5
n
,
the
only
parallelizable
ones
are
51,
53
and
57.
The
first
two
are
Lie
group
manifolds,
corresponding
to
the
group
U(l)
and
5U(2)
respectively. )
Bibliography
[1]
I.M. Singer
and
J.A. Thorpe: 'Lecture Notes on
Elementary
Topology
and
Geometry',
Springer (1976).
[2]
T.
Eguchi, P.B. Gilkey
and
A.J. Hanson, 'Gravitation, Gauge Theories
and
Differential Geometry',
Phys. Rep
66,
213 (1980).
[3]
C. Nash
and
S.
Sen, 'Topology
and
Geometry for Physicists',
Academic
Press (1988).
[4]
M.
Nakahara, 'Geometry, Topology
and
Physics',
Taylor and Francis
(2nd
edition,
2003).
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Part
II:
Group
Theory
and
Structure
and
Representations
of
Compact
Simple
Lie
Groups
and
Algebras
N.
Mukunda
Indian Institute
of
Science (retired)
Bangalore
560 012, India
+
Jawaharlal
Nehru
Fellow
This page intentionally left blankThis page intentionally left blank
Introduction
to
Part
II
The
aim
of
these
notes is
to
provide
the
reader
with
a concise account
of
ba-
sic aspects
of
groups
and
group
representations, in
the
form needed for
most
applications
to
physical problems. While
the
emphasis is
not
so
much
on
com-
pleteness
of
coverage
and
mathematical
rigour as
on
gaining familiarity
with
useful techniques,
it
is
hoped
that
these
notes would enable
the
reader
to
go
further
on
his
or
her
own
in
any
specialised aspects.
The
main
interest
will
be
in
Lie groups
and
Lie algebras;
compact
simple Lie algebras;
their
classification
and
representation
theory;
and
some specialised topics such as
the
use
of
Dynkin
diagrams, spinors, etc.
These
notes
are
intentionally
written
in
an
informal style,
with
problems
and
exercises occasionally inserted
to
help
the
reader
grasp
the
points
being
made.