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(This question has a physics motivation).

Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at least, common use of these structures in some context ?

EDIT 1 : There is an article of Edward Witten, Notes On Super Riemann Surfaces And Their Moduli,

EDIT 2 : There is a book, Supergravity, by Daniel Z. Freedman and Antoine Van Proeyen, Cambridge Editions.

Part IV title is "Complex Geometry and SUSY", where SUSY means supersymmetry. Kähler manifolds are used in this part.

Summary of the book :

Part I. Relativistic Field Theory in Minkowski Spacetime: 1. Scalar field theory and its symmetries; 2. The Dirac field; 3. Clifford algebras and spinors; 4. The Maxwell and Yang-Mills gauge fields; 5. The free Rarita-Schwinger field; 6. N=1 global supersymmetry in D=4; Part II. Differential Geometry and Gravity: 7. Differential geometry; 8. The first and second order formulations of general relativity; Part III. Basic Supergravity: 9. N=1 pure supergravity in 4 dimensions; 10. D=11 supergravity; 11. General gauge theory; 12. Survey of supergravities; Part IV. Complex Geometry and Global SUSY: 13. Complex manifolds; 14. General actions with N=1 supersymmetry; Part V. Superconformal Construction of Supergravity Theories: 15. Gravity as a conformal gauge theory; 16. The conformal approach to N=1 supergravity; 17. Construction of the matter-coupled N=1 supergravity; Part VI. N=1 Supergravity Actions and Applications: 18. The physical N=1 matter-coupled supergravity; 19. Applications of N=1 supergravity; Part VII. Extended N=2 Supergravity: 20. Construction of the matter-coupled N=2 supergravity; 21. The physical N=2 matter-coupled supergravity; Part VIII. Classical Solutions and the AdS/CFT Correspondence: 22. Classical solutions of gravity and supergravity; 23. The AdS/CFT correspondence;

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I think you should try to explain your motivation and what kind of relationship you would expect. As it stands, the question is very vague. – Henry T. Horton May 4 '13 at 22:45
This is true, but, because I don't know very well the subject, I am precisely looking for somebody who is able to shade a light about some relationship. One of the manifolds has to do with supersymmetry. The other kind of structure has to do with compatible Riemaniann, complex, symplectic structures. I was just wondering if there could be any relation about these 2 kind of structures, but maybe it is not the case. – Trimok May 5 '13 at 15:52
@Trimok: why do you think there should be a relationship? did someone mention this in a seminar? do you have some examples in which there is some strange relationship you'd like to generalize? I don't know any physics, so I don't think I can help at all, but I don't even understand where the question is coming from. – Sam Lisi May 7 '13 at 23:13
I made an edit,in the question, to signal a book – Trimok May 9 '13 at 8:33
@Trimok, The book of Van Proeyen ? He lectures me and I'm to make an assignment on chapter 13. If you might have found the answer, I'd like to hear it :). I could also have a talk with him after the discussion of the assignment :). – Dominique Jan 17 '14 at 12:39

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