Something I read somewhere about Spin manifolds, I don't remember where?

In summary, the conversation discusses the weights of SO(3) on a spin manifold and a regular manifold. It is mentioned that on a spin manifold, the weights of SO(3) take values in Z/2, while on a regular manifold, they take values in Z/4. The conversation ends with a request for someone to explain this further.
  • #1
Jim Kata
197
6
What I'm about to say is really sketchy since I don't even remember where I read this, and don't really understand the topic, but I thought it was cool. Basically, it said on a spin manifold, a manifold where can do a lift from, for example an SO(3) twisted bundle to Spin(3), the weights of SO(3) take their in values in Z/2. That is nothing new, but it said on a regular manifold, that is, I think, one where SO(3) is not the tangent bundle, SO(3) weights would have values in Z/4.

Can anyone elaborate?
 
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  • #2
I found what I read. It is not as interesting sounding as I remembered, but it says

On the other hand, for
SO(3), the instanton number takes values in Z/4, or in Z/2
if M is a spin manifold.

Can anyone explain this?
 

1. What are Spin manifolds?

Spin manifolds are a type of differential manifold, which is a geometric object that serves as a mathematical model for physical space. Spin manifolds have additional structure that allows for the definition of spinors, which are mathematical objects used to describe spin in quantum mechanics.

2. How are Spin manifolds different from other manifolds?

Spin manifolds are different from other manifolds because they have a spin structure, which is a mathematical construction that allows for the definition of spinors on the manifold. This structure is important in studying the behavior of particles with spin in quantum field theory.

3. What is the significance of Spin manifolds in physics?

Spin manifolds are significant in physics because they allow for the study of spinors, which are essential in describing the behavior of particles with spin. This is important in understanding the behavior of fundamental particles and their interactions.

4. Can Spin manifolds be visualized?

Spin manifolds themselves cannot be visualized, as they are abstract mathematical objects. However, their underlying geometry can be visualized in certain cases, such as in the case of a 2-dimensional sphere, which is a simple example of a Spin manifold.

5. Where can I find more information about Spin manifolds?

There are many resources available for learning about Spin manifolds, including textbooks, research articles, and online lectures. Some good starting points include "Spin Geometry" by Lawson and Michelsohn, and "Spinors and Space-Time" by Penrose and Rindler.

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