1. Gauge Fields, Knots and Gravity Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg matwml@nus.edu.sg (65)96314907 Lecture based on book (same title) by John Baez and Javier Muniain

2. Objective & Strategy Manifold, Tangent Space, Bundle, Vector Field Survey Entire Book – a Vast Landscape Maxwell, Yang-Mills, Chern-Simons, Knots Familiar Peak : Basic Math, Linear Algebra, Calculus View the Landscape from a Peak Lecture One: Structures, Affine Spaces, Derivatives Differential Form, Exterior Derivative, DeRham Theory Affine Connection, Covariant Derivative, Curvature General Relativity, ADM, New Variables Lie Groups, Lie Algebras, Flows, Principle Bundles

3. Structures Elements – propositional and predicate logic Products – relations (equivalence, order, functions) Functions Sets Composition – category of sets Structures – semigroups, groups, rings, fields, modules, vector spaces, algebras, Lie algebras Algebra

4. Vector Fields on Sphere Two Dimensional Sphere – A Manifold Module of Tangent Vector Fields Ring of Continuous Real-Valued Functions Free Module of Vector-Valued Continuous Functions Theorem This module is spanned by three elements, but it is not free (it does NOT have a basis).

5. TRANSFORMATION GROUPS a setis a group Stabilizer Subgroup and a function define OrbitFor Theoremis the set of left cosets Theorem Definitions Transitive and Free Transformation Groups

6. AFFINE SPACE is a vector space and a transformation group Example 1 that is both transitive and free, this means that Example 2 over a field

7. AFFINE SPACE in an affine space we can define sub-affine spaces, eg lines, planes, etc that correspond to orbits of subspaces of For any point however an affine space is NOT a vector space, however we can also define affine transformations of S by using translations of V and linear transformations of V

8. Bases and Charts definea basis We can parameterize an affine space S as follows: and construct a mapping Choose 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

9. EUCLIDEAN SPACE Is an quadruplet is a real affine space positive definite : bilinear : a linear function of each argument is a mapping that is symmetric : Definitionare orthogonal if Definition have distance Question Is this what Euclid had in mind ?

10. REAL AFFINE SPACE For finite dimensional V, a canonical topology on S, and a canonical differentiable structure on V is one where F = R, in such a space we can define Convex combinations of points in S If is differentiable (at t) then Its time to turn our attention to derivatives

11. A New Look at Derivatives and then there exists is linear and satisfies Theorem If 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

12. DERIVATIONS a point and more generally to any manifold (to come). and the concept can be extended to Definition Tangent Space at Definition Functions like L are called derivations at Definitions Vector fields as derivations on is the set of derivations atand denoted by Remarks Why tangent spaces on the sphere are different, Lie algebra of vector fields, Leibniz Law and binomial theorem, exponential of derivations

13. MANIFOLDS is Hausdorff, paracompact, and admits of charts Definition A Manifold is a topological space X that Implicit Function Theorem  Ifand and there exists such that thenis adimensional manifold.

14. MANIFOLDS

15. Tangent Space Category Contravariant Functor Covariant Functor

16. Tangent Space such that Recall that is linear and satisfies Continuous functions map tangents to tangents since

17. Fiber Bundles Definition Homeomorphisms (local trivializations) Fiber  Transition Functions satisfy where

18. Tangent Bundle Charts yield Definition 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

19. Tangent Bundle Example where chart 1 is given by stereographic projection If we identifythen Remark hence Chern Class = -2 and chart 2 is ster. proj. composed with y  -y

20. Induced Bundles Definition Given a bundle we construct the induced (or pullback) bundle and a continuous map and where http://planetmath.org/encyclopedia/InducedBundle.html

21. Sections Definition A section of a bundle is a continuous map Remark For each local trivialization of a bundle and choice ofwe can construct a section of the bundle that is induced by since

22. 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 satisfy defined by

23. 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 the vertical distribution d on E is defined by Definition For a smooth bundle (spaces manifolds, maps smooth) 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

24. Holonomy of a Connection Theorem Given a bundle 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. 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. 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.

25. 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.