Gauge Fields, Knots and Gravity

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Gauge Fields, Knots and Gravity

• Wayne Lawton
• Department of Mathematics National University of
  Singapore matwml_at_nus.edu.sg
• (65)96314907

Lecture based on book (same title) by John Baez
and Javier Muniain
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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
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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
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Vector Fields on Sphere

• Two Dimensional Sphere A Manifold

Ring of Continuous Real-Valued Functions
Free Module of Vector-Valued Continuous Functions
Module of Tangent Vector Fields
Theorem This module is spanned by three elements,
but it is not free (it does NOT have a basis).
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TRANSFORMATION GROUPS

a set
is a group
and a function
define Orbit
For
Stabilizer Subgroup
Theorem
is the set of left cosets
Theorem
Definitions Transitive and Free Transformation
Groups
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AFFINE SPACE

is a vector space
over a field
and a transformation group
that is both transitive and free, this means that
Example 1
Example 2
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AFFINE SPACE
For any point
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
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Bases and Charts
We can parameterize an affine space S as follows
define
a basis
Choose
and construct a mapping
by
where
If V is finite dimensional and B is an ordered
basis then
is a chart and its entries are the coordinate
functions on S
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EUCLIDEAN SPACE
Is an quadruplet
is a real affine space
is a mapping that is
bilinear a linear function of each argument
symmetric
positive definite
Definition
are orthogonal if
Definition
have distance
Question Is this what Euclid had in mind ?
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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
is differentiable (at t) then
If
Its time to turn our attention to derivatives
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A New Look at Derivatives
and
is linear
Theorem If
and satisfies
then there exists
such that
Proof First we observe that
Taylors Theorem implies that there exists
such that
therefore
and the result follows by choosing
Remark this is the converse of Leibniz Law
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DERIVATIONS
Definition Functions like L are called
derivations at
a point
and the concept can be extended to
and more generally to any manifold (to come).
Definition Tangent Space at
is the set
of derivations at
and denoted by
Definitions Vector fields as derivations on
Remarks Why tangent spaces on the sphere are
different, Lie algebra of vector fields, Leibniz
Law and binomial theorem, exponential of
derivations
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MANIFOLDS
Definition A Manifold is a topological space X
that
is Hausdorff, paracompact, and admits of charts
Implicit Function Theorem ? If
and
and there exists
such that
then
is a
dimensional manifold.
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MANIFOLDS
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Tangent Space
Category
Covariant Functor
Contravariant Functor
Covariant Functor
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Tangent Space
Recall that
such that
is linear and satisfies
Continuous functions map tangents to tangents
since
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Fiber Bundles
Definition
Fiber
Homeomorphisms (local trivializations)
? Transition Functions
satisfy
where
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Tangent Bundle
Definition
Charts
yield
where the fiber is homeomorphic to
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
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Tangent Bundle
Example
where chart 1 is given by stereographic
projection
and chart 2 is ster. proj. composed with y?-y
If we identify
then
Remark
hence Chern Class -2
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Induced Bundles
Definition Given a bundle
and a continuous map
we construct the induced (or pullback) bundle
where
and
http//planetmath.org/encyclopedia/InducedBundle.h
tml
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Sections
Definition A section of a bundle
is a continuous map
Remark For each local trivialization of a bundle
and choice of
we can construct a section
of the bundle that is induced by
since
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Vector Fields
Definition A vector field
on a manifold M is a section of the tangent
bundle, it corresponds to an R-transformation
groups (perhaps local) on M.
This means that the trajectories
defined by
satisfy
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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
Definition For a smooth bundle (spaces
manifolds, maps smooth)
the vertical distribution d on E is defined by
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
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Holonomy of a Connection
Theorem Given a bundle
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.
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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.