Talk:Graded manifold

Hi there. I do not know who created this entry but it is well written, although in a somewhat concise form. I was lead here by the comment that "this entry may be too technical". I have a paper trying to explain the rationale behind graded manifolds to physicists and non-expert mathematicians, that may be of some use. It is published in the "Advances in Mathematical Physics" journal: http://www.hindawi.com/journals/amp/2009/987524/ I would like to know the opinion of the entry's creator about if it would be a suitable reference. Also, I have a couple of technical comments: (1) The entry mentions Batchelor's theorem about the splitting of a supermanifold. I think some words should be written about the fact that this is true only for real supermanifolds, and fails for complex ones: P. Green: On holomorphic graded manifolds. Proc. of the American Math. Soc. 85 (1982) 587-590. Indeed, the existence of a splitting is equivalent to giving a graded Koszul connection, as proven by Koszul in J. L. Koszul: Connections and splittings of supermanifolds. Differential Geometry and its Applications 4, Issue 2 (1994) 151-161. (2) The article cites a paper by Antonio Almorox about the formulation of gauge theories on a graded manifold. That paper (which was the outcome of Antonio's Ph. D. thesis) is a little bit outdated, although it is a very good work that still maintains its interest as a mathematical exposition of the basic ground of graded manifold theory. The same can be said about the cited paper by Jaime Muñoz Masqué and Daniel Hernández Ruipérez. The point is that the Euler-Lagrange for field theories on a graded manifold that result from the direct application of the calculus of variations, do not give the equations used in Physics, because the natural integral on a supermanifold (the graded integral) is not the same as the Berezin integral. There is, however, a way to relate both formalisms, the Berezinian calculus of variations and the graded one, called the "comparison theorem", due mainly to J. Muñoz Masqué, J. Monterde and myself, which is an extension of these previous works by Jaime, Antonio and Daniel. Of course, I am aware that this sounds a little bit as "self-promotion", but here are a couple of references, just in case the creator of the page find them useful: J. Monterde and J. Muñoz Masqué: Variational problems on graded manifolds. Contemp. Math. 132 (1992) 551-572 J. Monterde, J. Muñoz-Masqué and J. A. Vallejo: The Poincaré-Cartan form in Superfield Theory. International Journal of Geometric Methods in Modern Physics 3 4 (2006) 775-822. Note that I have not modified anything. I am just making some suggestions. Cheers, J. A. Vallejo Josanv (talk) 22:21, 25 May 2012 (UTC)