Lectures on Geometrical Anatomy of Theoretical Physics by Frederic Schuller in 2015
Playlist of associated lecture videos
Notes by Simon Rea (see the books library in mendeley)
See also
The geometry of physics by Theodore Frankel in 2012
Physics and geometry by Edward Witten in 1986
L6. Topological manifolds and manifold bundles
3. Topological manifolds and bundles
- 3.1 Definition and construction of topological manifolds
-
3.2 Bundles
-
A bundle (of topological manifolds) is a triple \[(E, \pi, M)\]
- \[E\] is a topological manifold called the total space
- \[M\] is a topological manifold called the base space
- \[\pi : E \rightarrow M\] is a continuous surjective map from the total space onto the base space
- Let \[p \in M\], then \[\pi^{-1}(p) = F_p\] is called the fiber at \[p\]
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L7. Differentiable structures definition and classification
L19. Principal fiber bundles
left and right Lie group actions;
- example: actions from representations;
- proof: right actions from left actions;
equivariance of smooth maps;
orbits, orbit space and stabilisers;
free and transitive actions;
- examples;
smooth and principal bundles; (~53:00)
- detailed example: the frame bundle;
principal bundle morphisms and isomorphisms (or diffeomorphisms);
trivial bundles;
- proof that a bundle is trivial if and only if it admits a global section.
L20. Associated fiber bundles
associated fibre bundle to a principal bundle;
- detailed example: the frame bundle;
scalar and tensor densities on a manifold;
associated bundle maps and isomorphisms;
trivial associated bundles;
restrictions and extensions of a principal bundle;
examples.
L21. Connections and connection 1-forms
vertical and horizontal subspaces at a point;
decomposition in vertical and horizontal parts;
connection on a principal bundle;
connection one-form;
properties of connection one-forms with proof.
L22. Local representations of a connection on the base manifold: Yang-Mills fields
Topics covered
- Yang-Mills field as pull-back of a connection one form along a local section
- local trivializations of a principal bundle
- local representation of a connection one-form
- Maurer-Cartan form
- Example: the Yang-Mills fields on the frame bundle, Christoffel symbol
- Example: calculation of the Maurer-Cartan form of the general linear group \[GL(n, \mathbb{R})\]
- Patching Yang-Mills fields on different domains
- the gauge map
- Example: the gauge map on the frame bundle