Application: Spin structures - lec 27 - Frederic Schuller

The video begins with an introduction to the topic of spinner fields on curved spaces and spacetimes. It emphasizes the complexity of the subject, which requires a deep understanding of mathematical concepts such as principle bundles, associated bundles, and the formalism of spin geometry. The presenter mentions that the spin group is a double cover of the special orthogonal group, which is a key concept in defining spinner fields as sections of an appropriate bundle.

This paragraph delves into the properties of the spin group, explaining that it is a double cover of the special orthogonal group. The presenter discusses the homomorphism between the spin group and the orthogonal group, highlighting that the kernel of this homomorphism is isomorphic to a group with two elements. The dimensions of the spin group in various dimensions are explored, with a focus on three and four dimensions, which are of particular interest in physics.

The discussion shifts to the Lorentz group and the concept of a double cover in the context of the spin group. The presenter explores the spin groups in different dimensions, particularly in one, two, and three dimensions, and their corresponding groups. The paragraph also touches on the accidental isomorphisms that occur in certain dimensions and the importance of understanding these groups in the context of relativity.

The focus of this paragraph is on constructing a Lie group homomorphism from SU(2) to SO(3). The presenter outlines the need to work with the Lie algebra su(2) and to understand its structure. The process of finding a map that takes elements from the Lie algebra of SU(2) to those of SO(3) is discussed, emphasizing the importance of the map's kernel being isomorphic to Z2.

The video script explains the concept of the adjoint action of a Lie group and how it can be used to represent the group. The presenter discusses the Lie algebra su(2) and its representation through the adjoint action. The process of finding the tangent vectors at the identity of the group is described, leading to the conclusion that these vectors form a space of traceless Hermitian matrices.

This paragraph explores the isomorphism between the real vector space R3 and the space H of traceless Hermitian matrices. The presenter defines an isomorphism mu and its inverse, and demonstrates that mu is a linear map with an explicitly constructed inverse, confirming its status as an isomorphism. The paragraph also discusses the compatibility of this isomorphism with the inner product structures of R3 and H.

The concept of a spin frame bundle over an n-dimensional manifold is introduced. The presenter explains that this is a principal spin n bundle that maps to the orthogonal frame bundle. The conditions for a map to be a bundle morphism and to be Rho invariant are discussed. The paragraph also touches on the existence of a spin structure and its relation to the topology of the manifold.

The video script generalizes the construction of spin structures to spin(1, n-1) on Lorentzian manifolds. The presenter explains that the same construction can be applied to Lorentzian manifolds, which are like Riemannian ones but with a weakened positivity condition. The importance of understanding the spin structure in the context of general relativity is highlighted.

The final paragraph discusses the importance of spin structures in physics, particularly in defining spinner fields as sections of a spin bundle. The presenter explains that the concept of spin is more elementary than tensors and requires a metric geometry. The paragraph also clarifies that the understanding of spin relies on the existence of a metric or a Lorentzian structure, which is not necessary for tensors.