Transcript Slide 1

Gauge Fields, Knots and Gravity
Wayne Lawton
Department of Mathematics
National University of Singapore
[email protected]
(65)96314907
Lecture based on book (same title) by
John Baez and Javier Muniain
Objective & Strategy
Survey Entire Book – a Vast Landscape
Manifold, Tangent Space, Bundle, Vector Field
Differential Form, Exterior Derivative, DeRham Theory
Lie Groups, Lie Algebras, Flows, Principle Bundles
Affine Connection, Covariant Derivative, Curvature
Maxwell, Yang-Mills, Chern-Simons, Knots
General Relativity, ADM, New Variables
View the Landscape from a Peak
Familiar Peak : Basic Math, Linear Algebra, Calculus
Lecture One: Structures, Affine Spaces, Derivatives
Structures
Sets
Elements – propositional and predicate logic
Products – relations (equivalence, order, functions)
Functions
Composition – category of sets
Algebra
Structures – semigroups, groups, rings, fields,
modules, vector spaces, algebras, Lie algebras
Vector Fields on Sphere
Two Dimensional Sphere – A Manifold
S  {v  R |  v  1}
2
3
3
2
i 1 i
Ring of Continuous Real-Valued Functions
2
C ( S , R)
Free Module of Vector-Valued Continuous Functions
2
3
C (S , R )
Module of Tangent Vector Fields
Vect(S )  {F  C(S , R ) | F (v)  v, v  S }
2
2
3
2
Theorem This module is spanned by three elements,
but it is not free (it does NOT have a basis).
TRANSFORMATION GROUPS
G, a set S , and a function
T
G S 
 S  Tg : S  S | g G
is a group


 T1 ( x)  x, x  S  Ta  Tb  Tab , a, b  G
For
xS
define Orbit
Ox  Tg ( x) | g  G
S x   g  G | Tg ( x)  x 
Ox  G / S x is the set of left cosets
1
STg ( x )  g  S x  g
Stabilizer Subgroup
Theorem
Theorem
Definitions Transitive and Free Transformation Groups
AFFINE SPACE
is a vector space V over a field F
and a transformation group
V S 
 S
T
that is both transitive and free, this means that
p, q  S , !v V  Tv (q)  p
Example 1 S  V , T ((v, s))  v  s
Example 2 S  { p, q}, V  F  Z
2
T (0, p)  p, T (0, q)  q
T (1, p)  q, T (1, q)  p
AFFINE SPACE
p
v  pq
r  v  Tv ( r )
q
v
r
For any point
x  S , S  Ox  V /{0}  V
however an affine space is NOT a vector space, however
in an affine space we can define sub-affine spaces, eg
lines, planes, etc that correspond to orbits of subspaces of
we can also define affine transformations of S by using
translations of V and linear transformations of V
V
Bases and Charts
We can parameterize an affine space S as follows:
Choose
xS ,
a basis
B V,
define
F0 ( B)  { f : B  F | f finitesupport}
and construct a mapping  : S  F ( B) by
0
 ( y)  f  F0 ( B), y  S
where
y  x  bB f (b)b
If V is finite dimensional and B is an ordered basis then

F0 ( B)  F
dim(V )
is a chart and its entries are the coordinate functions on S
EUCLIDEAN SPACE
(S, V, T ,  )
(S, V, T ) is a real affine space
 :V V  R is a mapping that is
Is an quadruplet
bilinear : a linear function of each argument
 (u, v)   (v, u)
positive definite u  0,  (u, u )  0
:
Definition u, v V are orthogonal if  (u, v)  0
symmetric :
Definition
p, q  S have distance
d ( p, q)   ( p  q, p  q)
Question Is this what Euclid had in mind ?
REAL AFFINE SPACE
is one where F = R, in such a space we can define
Convex combinations of points in S
For finite dimensional V, a canonical topology on S,
and a canonical differentiable structure on V
If
:RS
is differentiable (at t) then
 (t )  lim  ( (t   )   (t )) V
1
 0
Its time to turn our attention to derivatives
A New Look at Derivatives

Theorem If a  R and L : C ( R)  R is linear
L( fh)  f (a) L(h)  L( f )h(a)
then there exists c(a)  R such that

L( f )  c(a) ( Df )(a), f  C ( R)
Proof First we observe that L(constant)  0

Taylors Theorem implies that there exists h  C (R)
such that f ( x)  f (a)  ( x  a)h( x), x  R
therefore Lf  L( x)h(a)  L( x)(Df )(a)
and the result follows by choosing c(a)  L( x)
and satisfies
Remark : this is the converse of Leibniz Law
DERIVATIONS
Definition Functions like L are called derivations at
a point
a  R and the concept can be extended to

L : C (S )  R
and more generally to any manifold (to come).
Definition Tangent Space at
a
aS
is the set
Ta (S )

Definitions Vector fields as derivations on C (S )
of derivations at
and denoted by
Remarks Why tangent spaces on the sphere are
different, Lie algebra of vector fields, Leibniz Law
and binomial theorem, exponential of derivations
MANIFOLDS
Definition A Manifold is a topological space X that
is Hausdorff, paracompact, and admits of charts

 U  X  U  R
d
  1
  (U   U  )   (U   U  )
R 
 R and
m
X  p  R | F ( p)  0 and there exists
k  m such that rank( DF ( p))  k , p  X
then X is a m  k dimensional manifold.
Implicit Function Theorem  If

m

F
n
MANIFOLDS
R  S  {( x, y) | x  y  1}
2
1
2
2
R  S  {( x, y, z) | x  y  z  1}
3
2
2
2
2
R  S  S  {( x, y, z) | F ( x, y, z)  0}  R
4
1
1
F ( x, y , z ) 

1
4
(x  y  z ) 
2
2
2
33
3
4

2
3
x y
2
SO(3, R)  {M  R : F (M )  0}
T
10
F (M )  ( I  M M ,1  det(M ))  R
2
Tangent Space
( X , p) 
(Y , q) f ( p)  q
f
Covariant Functor X 
 Y *
f


Contravariant Functor C (Y ) 
C ( X )
*
f (h)  h  f
f
Category
Covariant Functor
Tp ( X ) Tq (Y )
f*

( f* (v p ))q (h)  v p ( f (h)), v p Tp ( X ), h  C (Y )
*
cont  cov  cov  cont  cont, cont  contr  cov
1 *
d
q U   X , Tq ( X )   T ( q ) ( R )
d
d
T ( q ) ( R )  { i 1 v i  xi }
 

Tangent Space
vp

Recall that T ( X )  {C ( X ) 
 R}
p
such that v p is linear and satisfies

v p (ab)  a( p)v p (b)  v p ( p)b(a), a, b  C ( X )
Continuous functions map tangents to tangents since

( f* (v p ))q (ab)  v p ( f (ab)), v p Tp ( X ), h C (Y )
*
f (ab))  (ab)  f  (a  f )(b  f )  ( f a)( f b)
*
*
*
v p ( f (ab))  f (a)( p)v p ( f (b))  v p ( f (a)) f (b)( p)
*
*
*
 a(q)( f* (v p ))q (b)  ( f* (v p ))q (a)b(q)
*
*
Fiber Bundles

Definition E 
 B, B   U
1
Fiber  (u)  F , u  B

1

(
U
)


U

F


Homeomorphisms
(local trivializations)   1 (u, p)  u

 Transition Functions
  1
(U  U  )  F  
(U   U  )  F
1
satisfy    (u, p)  (u, g (u)( p))



where g (u)  1, u U


1
g (u)  ( g (u)) , u U U 
g (u) g (u)  g (u), u U U  U
Tangent Bundle
Definition
Charts
yield

T ( X )   pX Tp ( X ) 
 X

X  U  R
d
  ( )*
 (U ) U  R
1
where the fiber is homeomorphic to
R
d
d
Problem Show that the transition functions are
linear maps on each fiber and derive explicit
expressions for them in terms of the standard
coordinates on R d
Example
Tangent Bundle
2
2
S  U1 U 2 U k  S \ { pk }
where chart 1 is given by stereographic projection
k
2
Uk  R  C  {x  iy | x, y  R}
and chart 2 is ster. proj. composed with y-y
T (U k )    T (C) then
1
1


d 1
1  2 ( z, (a x  b y )  ( z , ( dz z )(v))


z  C \ {0}, v  a  ib  a x  b y Tz (C)
Remark z 1 (v)  d z 1 v   z 2v
*
dz
1
k *
If we identify
 
hence Chern Class = -2


Induced Bundles

Definition Given a bundle E 
 B
f
and a continuous map B ' 
 B
we construct the induced (or pullback) bundle
f * ( )
'
'
E  B
where
E  { (e, b)  E  B | f (b)   (e) }
'
and
'
f ( )((e, b))  b
*
http://planetmath.org/encyclopedia/InducedBundle.html
Sections
E
 B
s
B
 E    s  id B
Definition A section of a bundle
is a continuous map

Remark For each local trivialization of a bundle
 (U ) U  F
1

p  F we can construct a section s p
i
of the bundle that is induced by U 
  B
'
E  { (e, u)  E U | u   (e) }
and choice of
s p (u)   ((u, p)) since    (u, p)  u
1
1
Vector Fields
v
Definition A vector field M 
T (M )
on a manifold M is a section of the tangent bundle, it
corresponds to an R-transformation groups (perhaps
local) on M.
Tr : M  M , r  R
gp
This means that the trajectories R  M
defined by g p (r )  Tr ( p),
pM
satisfy
g  ((r,
p *

r
))  v( g p (r)) Tg p ( r ) (M ), r  R
Distributions and Connections
Definition A distribution d on a manifold M is a map
that assigns each point p in M to subspace of the
tangent space to M at p so the map is smooth. Two
distributions c and d are complementary if
Tp ( E)  c( p)  d ( p), p  E

Definition For a smooth bundle
E


B
(spaces manifolds, maps smooth)
the vertical distribution d on E is defined by
d ( p)  { v Tp (M ) |  * (v)  0 }
Definition A connection on the bundle is a distribution
on E that is complementary with the vertical distribution
Theorem For a connection c the projection induces an
induces isomorphism of c(p) onto T_p(B) for all p in E
Holonomy of a Connection

Theorem Given a bundle E 
 B
and points p, q in B then every nice path (equivalence
set of maps from [0,1] into B) defines a diffeomorphism
(holonomy) of the fiber over p onto the fiber over q.
Proof Step 1. Show that a connection allows vectors
in T(B) be lifted to tangent vectors in
T(E) Step 2. Use the induced bundle construction to
create a vector field on the total space of the bundle
induced by a map from [0,1] into B. Step 3. Use the
flow on this total space to lift the map. Use the lifted
map to construct the holonomy.
Remark. If p = q then we obtain holonomy groups.
Connections can be restricted to satisfy additional
(symmetry) properties for special types (vector,
principle) of bundles.
Theorem of Frobenius
Theorem A distribution is defined by a foliation iff
it is involutive.This means that if v and w are two
vector fields subordinate to the distribution then their
commutator [v,w] is also subordinate to the
distribution. All involutive distributions give trivial
holonomy groups if the base manifold is simply
connected.