Reference request: Supermanifolds with open charts approach

In summary, the speaker is looking for a book on supermanifolds for their undergraduate thesis on supersymmetry. They found a book by famous mathematicians and physicists but are struggling with the use of sheaves and schemes. They are asking for recommendations on books that cover general topics on supersymmetry without using sheaves and schemes. Two possible references are suggested by the respondent.
  • #1
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I've decided to work on some topics about supersymmetry for my undergraduate thesis. My background is that I have had a one-semester course in differentiable manifolds, plus typical undergrad math courses, real analysis and module theory (graduate algebra).

So, I went to the library and found a book written by some famous mathematicians and physicists like Witten and Delign. It's called Quantum Fields and Strings: A course for mathematicians. I started reading the first two chapter. My main problem with the book is that it defines supermanifolds by sheaf of algebras and I don't know anything about sheaves or schemes and I don't have time to study algebraic geometry this semester. Is there a good book that covers general topics that students need to know about supersymmetry without using sheaves and schemes?

What books do you think I should look at? Thanks in advance.
 
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  • #2
If I understand your question right, you are looking for a reference about super manifolds, and not necessarily about supersymmetry in quantum field theory, which is what people usually mean when they say supersymmetry? If so, you can try to have a look at
http://arxiv.org/abs/1209.2199
if you are interested in supersymmetric quantum field theories, have a look at
http://arxiv.org/abs/hep-ph/9709356
 

1. What are supermanifolds and how are they different from regular manifolds?

Supermanifolds are mathematical objects that extend the concept of a manifold, which is a space that can be locally approximated by Euclidean space. Supermanifolds incorporate the idea of supersymmetry, a symmetry between bosons (particles with integer spin) and fermions (particles with half-integer spin). This leads to the inclusion of anticommuting coordinates in addition to the usual commuting coordinates found in regular manifolds.

2. What is the open charts approach to studying supermanifolds?

The open charts approach is a method of describing supermanifolds by using open subsets (or "charts") of the manifold that cover the entire space. In this approach, the supermanifold is represented as a collection of local charts, each of which is isomorphic to a patch of Euclidean space. This allows for a more concrete and manageable description of supermanifolds compared to other approaches.

3. Why is the open charts approach useful for studying supermanifolds?

The open charts approach simplifies the study of supermanifolds by breaking them down into smaller, more manageable pieces. This allows for a better understanding of the structure and properties of supermanifolds, as well as making calculations and proofs easier to handle. It also provides a more intuitive way of visualizing and working with supermanifolds.

4. Are there any applications of supermanifolds in physics or other fields?

Yes, supermanifolds have been used in various areas of physics, including string theory, quantum field theory, and supersymmetry. They have also found applications in mathematics, such as in the study of supergeometry and superalgebras. Additionally, the open charts approach has been used to develop new techniques for solving differential equations and studying geometric structures.

5. Where can I find more information about supermanifolds with the open charts approach?

There are many resources available for learning about supermanifolds and the open charts approach. Some recommended sources include textbooks such as "Supermanifolds: Theory and Applications" by Alice Rogers and "Supersymmetry and Supergravity" by Julius Wess and Jonathan Bagger. Online resources, such as lecture notes and research papers, are also available for further reading and study.

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