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May 26, 2005

Talking to myself

Posted by Robert H.

Since nobody answered my question about how to properly generalize the calibration condition for BPS-branes if the gauge field has curvature I have to do it myself.

Until this very minute, I have been preparing for today’s joint math/physics block seminar in Hamburg where we’re going to find out what Generalized Complex Geomtery is really about (Lubos has chatted about it in his reference frame). Not to arrive completely clueless I have been reading Gualtieri’s thesis which I can strongly recommend to everybody. It is an excelent read even for physicists! And there, in chapter 7 my question is answered:

You probably know that this generalized business works by considering the tangent and co-tangent bundles together. Then a generalized complex structure JJ is maps TT *T\oplus T^* to itself and squares to 1-1 and fulfills some integrability condition. It’s easy to see that this condition contains complex, symplectic and Poisson geometry and interpolates between these. Furthermore it is co/invariant under transformations by closed 2-forms BB and can be twisted by closed threeforms HH, e.g. dBdB for not closed BB.

Now consider a submanifold of this space on which there is a 2-form FF with dF=HdF=H (0 without twist). The trick now is to look at the subbundle of TT *T\oplus T^* on the submanifold such that the vector component XX is tangent to the submanifold and the form component is given by i XFi_X F. The condition for this to be a generalized complex submanifold is now to require that this bundle is stable under JJ. And, as promised, this generalizes complex, Lagrangian and self-duality for FF as BPS conditions. And there is also a spinorial description.

I must say, this story is one of those that is so beautiful that it can really foster your belief that there must be some truth to string theory!

Posted at 8:23 AM UTC | Permalink | Followups (2)

May 18, 2005

String(n), Part II

Posted by Urs Schreiber

In the last entry I have listed some facts related to the group String(n)\mathrm{String}(n). Here is the literature that my discussion was mainly based on as well as a review of what String(n)\mathrm{String}(n) has to do with 2-groups and 2-bundles.

Posted at 1:17 PM UTC | Permalink | Followups (3)

String(n), Part I

Posted by Urs Schreiber

I was asked to say something about the meaning of the group String(n)\mathrm{String}(n) and about manifolds with string structure.

So here I’ll try to give a somewhat more comprehensive discussion than last time that we talked about this.

Posted at 9:58 AM UTC | Permalink | Followups (19)

May 17, 2005

Nonabelian Weak Deligne Hypercohomology

Posted by Urs Schreiber

What I described last time is really best thought of in the context of what I propose to call nonabelian weak Deligne hypercohomolgy.

Unless I am hallucinating the following is the correct formalism to generalize the well-known Deligne hypercohomology formulation of strict abelian pp-gerbes to weak and nonabelian pp-gerbes.

Posted at 1:46 PM UTC | Permalink | Post a Comment

May 16, 2005

PSM and Algebroids, Part V

Posted by Urs Schreiber

[Update: The following has become section 13 of hep-th/0509163.]

Last time I discussed how the functorial definition of a pp-bundle with pp-connection can locally be differentiated to yield morphisms between pp-algebroids. Now I think I have figured out the differential version of the transition law describing the transformation of these algebroid morphisms from one patch to the other. The result is a formalism that allows you to derive the infinitesimal cocylce relations of a nonabelian pp-gerbe with curving and connection, etc. using just a couple of elementary steps.

Posted at 4:05 PM UTC | Permalink | Followups (6)

May 13, 2005

PSM and Algebroids, Part IV

Posted by Urs Schreiber

Last time I discussed how Lie pp-algebroids (and hence Lie pp-algebras) and dg-algebras on graded vector spaces of maximal grade pp are two aspects of the same thing. This goes a long way towards merging the study of pp-bundles with pp-connections with the study of algebroid morphisms as they arise in the Poisson σ\sigma-model, Dirac σ\sigma-models and other field theories.

Here are more details.

Posted at 4:50 PM UTC | Permalink | Followups (1)

May 4, 2005

PSM and Algebroids, Part III

Posted by Urs Schreiber

I have just returned from visiting Thomas Strobl at Jena University, where we talked about algebroids, gerbes, categorified gauge theory, and generalized geometry and how it all fits together. I have learned a lot in these discussions and have gotten a little closer to seeing the big picture, also thanks to the valuable pointers to the literature by Melchior Grützmann and Branislav Jurčo. Here I’ll list some useful and interesting facts – except for those that are top-secret…

(Please note that all my attributions in the following reflect only my level of awareness of the literature. I’d be grateful for corrections and further pointers to the literature.)

[Note: Users of non-Mac machines might have to download a new font in order to properly view all mathematical symbols in the following. More general information can be found here.]

Posted at 9:35 AM UTC | Permalink | Followups (6)