Japanese version I translated into Japanese the article titled "generalized complex structure" on en.wiki. Because there are no article titled so on Japanese version. The original is the following:
1, As comment in the head of the article in en.wiki, the following arxiv: by Gukov, Dijikgraaf, Neitzke and Vafa was published.
Topological M-theory as Unification of Form Theories of Gravity hep-th/0411073
Hitchin functional plays a essential role in it, while generalized complex structure are considered by Professor Hitchin and his students.
2, As my comment in Japanese version, in 2001 Kapustin and Orlov had pointed out "coisotropic A-branes." And Gualtieri had related it to generalized complex structure in his paper. (It was after this article on en.wiki.)
By Gualitieri arXiv:math/0401221, which title is "Generalized complex geometry," has been refined as arXiv:math/0703298.
In this preprint He refers
"A. Kapustin and D. Orlov. Remarks on A-branes, mirror symmetry, and the Fukaya category J. Geom. Phys., 48:84, 2003, hep-th/0109098,"
which had more earlily proposed about coisotropic A-branes,
and said about chapter 6 the followings.
Finally, in part 6, we introduce generalized complex branes, which are vector bundles supported on submanifolds for which the pullback of the ambient gerbe is trivializable, together with a natural requirement of compatibility with the generalized complex structure. The definition is similar to that of a D-brane in physics; indeed, we show that for a usual symplectic manifold, generalized complex branes consist not only of flat vector bundles supported on Lagrangian submanifolds, but also certain bundles over a class of coisotropic submanifolds equipped with a holomorphic symplectic structure transverse to their characteristic foliation. These are precisely the co-isotropic A-branes discovered by Kapustin and Orlov [22].
1, As comment in the head of the article in en.wiki, the following arxiv: by Gukov, Dijikgraaf, Neitzke and Vafa was published.
ReplyDeleteTopological M-theory as Unification
of Form Theories of Gravity hep-th/0411073
Hitchin functional plays a essential role in it, while generalized complex structure are considered by Professor Hitchin and his students.
2, As my comment in Japanese version, in 2001 Kapustin and Orlov had pointed out "coisotropic A-branes." And Gualtieri had related it to generalized complex structure in his paper. (It was after this article on en.wiki.)
Gualtieri arXiv:math/0703298 Generalized Complex Structure
3, I suppose that they are greatly affected to the 6-dimensional quantum field theory by E. Witten. He used Hitchin functionals.
I posted to My BLOG in 9 May 2010 but in Japanese.
In the old comment above at ★２ The 6-th chapter of Mr. Gualitieri's Paper is described about coisotropic A branes. And I add it as a remark.
ReplyDeleteI stress that not only physicists but mathematicians (DG, the students of Prof. Hitchin) support coisotropic A branes by Kapustin and Orlov.
By Gualitieri arXiv:math/0401221, which title is "Generalized complex geometry," has been refined as arXiv:math/0703298.
ReplyDeleteIn this preprint He refers
"A. Kapustin and D. Orlov. Remarks on A-branes, mirror symmetry, and the Fukaya category
J. Geom. Phys., 48:84, 2003, hep-th/0109098,"
which had more earlily proposed about coisotropic A-branes,
and said about chapter 6 the followings.
Finally, in part 6, we introduce generalized complex branes, which are vector bundles supported
on submanifolds for which the pullback of the ambient gerbe is trivializable, together
with a natural requirement of compatibility with the generalized complex structure. The definition
is similar to that of a D-brane in physics; indeed, we show that for a usual symplectic
manifold, generalized complex branes consist not only of flat vector bundles supported on
Lagrangian submanifolds, but also certain bundles over a class of coisotropic submanifolds
equipped with a holomorphic symplectic structure transverse to their characteristic foliation.
These are precisely the co-isotropic A-branes discovered by Kapustin and Orlov [22].